NASA CONTRACTOR
REPORT
NASA CR-124075
REVISION
A
ISOGRID DESIGN HANDBOOK
McDonnell
5301
Botsa
Douglas
Astronautics
Company
Avenue
Huntington
Beach,
February
1973
[NASA-CR-12_075)
(McDonnell-Douqlas
222
p HC $13.25
Ca
92647
150GRID
DESIGN
Astronautics
U N73-19911
HANDBOOK
Co.)
CSCL
13M
G3/32
Prepared
For
NASA-GEORGE
Marshatt
Unclas
17333
Space
C.
Fright
MARSHALL
Centers
Alabama
SPACE
35812
FLIGHT
CENTER
ISC, GRID
MCDONIVELL_
D¢.)UGI.AS
DESIGN
HANDBOOK
_
FEBRUARY
PREPARED
MDC G4295A
1973
BY
,V1CDONNELL
DOUGLAS
ASTRONAUFIL;S
CO_.,1PANY
FOR:
NASA
MARSHALL
CONTRACT
MCDONNELL
DOUGLAS
NAS
SPACE
FLIGItT
CEN]ER
8 28619
ASTRONAUTICS
COMPANY-WEmT
..........
_..4_ ¸
FOREWORD
This
program,
the
conducted
by
the
Beach,
Cal.;fornia
administered
development
of
McDonnell
under
the
isogrid
Douglas
design
handbook,
Astronautics
und_,r
NASA
Contract
the
direction
of
Company
NAS
John
8-Z8619.
Key,
was
at
The
Marshall
Huntington
contract
was
Space
Flight
Center,
the
direction
of
with
M.
Harmon
NASA.
The
I_cDonnell
Douglas
Dr.
George
Moe,
acting
as
principal
contributor
theory
to
investigator.
this
document,
being
P.
Harwood
and
information
analysis,
in
test,
and
Development
the
Dr.
techniques.
The
was
Research
analytical
O.
by
Direct:or,
and
Mr.
and
program
the
Mr.
Development,
Robert
R.
for
major
was
(2)
a
obtained
from:
phase
B
space
Space
Flight
Center,
(3)
funded
by
NASA
Marshall
Space
Flight
Center,
isogrid
structural
Appreciation
is
Flight
Center
isogrid
to
for
aerospace
funded
expressed
to
his
continued
by
Mr.
principal
2 and
4,
basic
include
the
Jack
(1)
studies
Marshall
tests
the
Sections
contributors
NASA
the
was
B.
Orlando.
manufacturing
progcams,
Meyer
responsible
I.
document
under
and
Other
J.
advanced
conducted
NASA
interest
in
structures.
il
the
results
of
Independent
shuttle
booster
an
isogrid
and
Goddard
Furman
the
of
(4)
Space
the
development
NASA
of
Research
study
funded
tank
test
program
the
Delta
program
Flight
Center,
Marshall
and
application
Space
of
CONTENTS
Section 1
Section 2
1. 1
1.2
Background
Use
of the
BASIC
THEORY
Section
3
4
001
1.0.001
1. O. 003
tlandbook
2.0.
001
2.0.001
2.3
Hooke's
Law
for
Isogrid
Rib-Grid
Extensional
and
Bending
Stiffness
for
Composite
Rib-Grid
and
Skin
Constructions
NonDimensional
Stiffnessesfor
2.4
Unflanged
Membrane
Isogrid
Stresses
2.5
2.6
Equivalent
Summary
of
2. O. 010
2. '3. OlI
2.0.016
2. 0. 01'9
2. 1
2.2
Section
1.0.
I£NT RO DU CT ION
Monocoque
Basic
ISOGRID
CHARACTERISTICS
ADVANTAGES
ANALY
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
TICAI.
Spherical
Cylinders
Cylinders
Cylinder
Pressure
ln-Plane
Infinite
In-Plane
of Sheet
Cutout
E::
Theory
and
2.0.
t:::
AND
3.0.001
TECttXIQUES
4.
Cap
with
t/eversed
Pressure
in Compression,
Bending
Under
Torsional
Shear
Under
Uniform
External
4. 1.001
4.2.001
4.3.001
1.001
-i. 4.
Concentrated
Sheet
Ix}ad
in
Concentrated
Load
at
4.5.001
Reinforcement
Optimum
Iso_rid
ill
001
Edge
Open
Isogrid
Shear
Webs
()pen
Isogrid
Cylinders
in
Compression,
Bending
Open
and
Skinned
Isogrid
tqates
Minimum
Overall
Weight
for
Cylinder
Subjected
to Axial
Compression
and
Bending
Note
on Use
of x, y; o,,
(.Hrves
Off-
004
4.6.001
4. 7. 001
4.8.001
4.
4.
q. 001
10. OOl
4.
a
11.001
12. 00l
I'.001
Section
5
NODAL
Section
6
TESTING
GEOMETRY
5.0.001
6.0.001
,?
N_L L_P_[
6.1
6.2
Section 7
Model Tests
Sub-Scale and Full-Scale
MANUFACTUR
_
TECHNIQUES
7. I
Introduction
7. Z
7.3
7.4
Machining
Forming
Non- Destructive
Inspection
Manufacturing
Acceptance
ADDENDUM
m----
ING
Tests
6.0.00]
6.0.002
7.0.001
7.0.001
7.0.001
7. O. 005
for
7.0.013
FIGURES
?
Relative
I. O. 003
1-1
S-IVB
1-2
Non-Standard
2-1
Plot
3-1
Isogrid
3-2
Distributing
3-3
Equipment
3-4
Skylab
3-5
Reinforced
3-6
Fail-
3-7
Access
Door
3-8
Speed
Brakes
3-9
Manufacturing
3-10
Forming
Isogrid
Sample
4.1-1
Optimum
Reverse
Curves
Pressure
for
Installation
of
_3(a,6)
is
of
2. O. 012
to
Attached
and
Hole
3.0.001
Analyze
Concentrated
Safe
i. O. 004
Equipment
Curves
Simple
Floor
Design
Costs
to
Wall
for
3.0.003
Load
Nodes
3
Grid
30.
005
30.
OO6
Concentrated
Load
Concept
Designs
Samples
of
Isogrid
of
Isogrid
Spherical
Spherical
Cap
4.2-2
Interaction
Showing
4.2-3
Weight
Isogrid
of Optimized
Cylinders
4.2-4
Design
of
4.3-1
X,
Y, a,
Torsional
6
Curves
Shear
0. 007
3
0.008
3
O. 008
3.
O. 010
with
Bulkheads
of Theoretical
Bending
Buckling
Isogrid
3
3.0.011
Length
Dependence
Compression
and
Curve
0.004
the
Compression-
4.
1. 010
4.
1.011
Axial
Solutions
4. 2. 003
of
4. 2. 004
Effect
Length
Critical
Cylinders
for
Cylinders
Under
4.3.
008
4.3-2
Master
Curves
for
Torsional
Shear
4.3.009
4.3-3
Master
Curves
for
Torsional
Shear
4.3.010
4.4-1
X,
Y,
Uniform
a,
5 Curves
External
for
Cylinders
Pressure
Under
4.4.010
4.4-2
Master
Curves
for
Uniform
External
Pressure
4.4.013
4.4-3
Master
Curves
for
Uniform
External
Pressure
4.4.014
4.5-1
Concentrated
4.8-1
Shear
4.
10-1
k 1,
a
4.
10-2
k2,
a,
5 ,.
4.
13-1
X,
Y,
a,
Load
4.
5. 002
4.
8.003
Curves
4.
10.
007
,-es
4.
10.
008
Curves
4.
13.
OO4
Panel
, 6
6
5.0.001
5-I
Isogrid
7.2-1
Machining
Isogrid
with
Unflanged
7.2-2
Machining
Isogrid
with
Flanged
7.2-3
A
7.2-4
Electrical
Discharge
(ECM)
Machining
and
Weight
Node
Tank
Wall
7.0.
Ribs
7. 0.003
Ribs
7. O. 004
Configuration
to
002
(EDM)or
Electrochemical
Reduce
Isogrid
Nodal
Size
7.0.005
7.3-1
42-Foot-Long
7.3-2
Hand
Straightening
7.3-3
Test
Cylinder
7.3-4
Thor
Delta
Isogrid
7.3-5
Thor
Delta
Test
Cylinder
7.3-6
Isogrid
in
Creep
Forming
7.3-7
Contour
Changes
Brake
7. O. 006
Press
Ribs
Geometry
Using
0.008
7,
0.009
7.
0,009
7.0.010
Fixture
Cones
v!
7.
7,
O. 012
7.0.014
TABLES
1-1
Isogrid
1.0.005
4-1
_a.
4.
I
vii
1.020
ISOGRID
thickness
of
skin
b
=
width
of
rib
d
=
depth
of
web
c
=
depth
of
flange
w
=
width
of
flange
s
=
t+d
h
:
height
a
=
leg
of
:=
m
d
t'
× = _'
=
web
plate
of
DEFINITIONS
thickness
of
unflanged
isogrid
triangle
triangle,
distance
i.e.,
c
_-
center
bd
wc
W' _ :
t-ff
non-dimensional
to
center
of
nodes
parameters
(l +_ +_I [3(I+6_
z +-3_I*×)z + I + _62 * _×z]
[32
3 [(1+6)-b_(l+k)]
bending
2
stiffness
(For
parameter.
unflanged
X:_t
isogrid,
_z : [3_(1+6)z + (1+_)(1+_62)].)
Et
K
2 (1
1
-
:
+ a +_)
extensional
stiffness
(Iz = 0 for
unflanged
stiffness
(g
unflanged
v
isogrid)
D
=(Et312
)(
(l.v2)
bending
: 0 for
1 + a +
isogrid)
:
teff
t (I + a + _)
(1_ = 0 for
T
t (1
+ 3a
unflanged
:
equivalent
unflanged
÷ 3t_)
thickness
for
membrane
stresses
isogrid)
=
equivalent
weight
isogrid)
vlll
thickness
(_
: 0 for
-.
O,
_L
t
01
t 1 + _ + _
Equivalent
= to obtain
E
_
=
E (1 + a + _)Z
thickness
correct
and
K and
Young's
D
modulus
(_ = 0 for
unflanged
isogrid)
Use of E* and t* in monocoque
resultants,
couples,
strains,
displacements.
equations
curvature
Ix
gives
changes
correct
and
stress
Section
IN TR ODU
1.
1
establishment
concepts
the
of
for
new,
aerospace
lightweight,
economical,
structures
has
long
and
been
an
efficient
objective
structural
of
NASA
and
industry.
Lightweight,
compression-load-carrying
craft,
booster,
DC-7
used
mechanically
which
are
of
were
designed
and
to
space
structures.
stringer,
stage,
an
well
as
the
and
45-degree
S-IVB
45
degrees.
The
0-
the
Saturn
vehi,:le
they
are
inherently
four
skin
and
as
result
have
little
in-plane
In
1964,
Dr.
Robert
R.
Meyer
under
set
out
the
optimum
90-def;ree
to
a
find
are
domes.
A
goal
was
comings
o,
the
0-
other
penalties
the
most
patterr
efficient
cylinders
to
such
promising
took
advantage
structure.
as
find
a
and
as
increased
was
triangulation
Independent
the
work
from
a
pattern
for
of
simple
aircraft
machined
pattern.
The
the
fact
Research
and
1.0.001
stages
capability.
the
integral
Reference
compressively
loaded
that
negated
without
the
was
short-
found
members.
Development
1-1,
introducing
that
promise
of
However,
contract,
triangular
significant
the
resistance
stiffening
show,,'d
in
through
by
concept
that
rotated
collapsing
patterns
weight.
con-
regimes.
arrangement
45-degree
leakage
load
NASA-MSFC
or
duplicated
u_ed
certain
DC..6
however,
patterns
torsional
structural
90-degree
of
in
prevented
stiffening
This
an
lin,<s
construction,
of
patterns
efficient
bar
skin
stage
square
stiffening
extremely
to
used
air-
the
Boosters,
second
all
as
constant-height
Thor,
of
such
because
S-II
integral,
The
to
as
with
the
part
anct
structures
vehicle,
pat=erns
frame,
structures.
stiffenec}
Saturn
form
Aircraft
stiffened
int(:graliy
the
90-degree
vehicle
9'l-degree
as
In
structures
attached
course
siderations.
to
C TION
BACKGROUND
The
0-
1
trusses
and
program.
to
This
are
w, ry
was
extended
After
be
many
years
of
development,
structure
for
shrouds,
orbital
The
In
new
a
phase
carrying
high
an
high
point
be
Orbital
is
"Isogrid"
caUed
vehicle
(1)
high
torsional
used
as
attach
points
(3)
capability
stiffening
of
members.
the
the
and
previously
the
It
is
is
associated
important
to
note
with
of
cost
pattern.
The
45-degree
waffle
in
The
the
cost
same
structural
in
ure
Evidently,
selection
of
of
material.
space
used
the
in
the
capability
of
attach
points
need
to
resist
very
The
isogrid
con-
nodal
points,
which
protection
system
3tandoff
the
attach
loads
orbiter
added
internal
conducted
to
vehicle.
These
that
having
the
fewest
parts.
Saturn
S-IVB
tank
usable
for
equipment
to
cylinder
panels
for
must
with
compression
supplement
tests
test
served
primary
much
fully
aft
Relative
costs
Figure
1-1,
with
to
tightness,
not
skirt
and
interstage,
to
mainly
in
the
compartment
former
functions
an
economical
1.0.002
cannot
The
with
realized.
are
a
to
built
installation
(see
of
Fig°
in
the
suggests
of
to
save
both
the
evidence
way
of
indicative
Oe ignored
pattern
cost
integrally
attributed
isogrid,
is
structural
leak
structure.
as
lowest
be
secondary
attachment
be
the
the
designed
assure
between
these
stage,
was
to
fashion
structure,
is
Delta
the
many
a few
was
Delta
recoverable
multiple
shown
style,
a concept
type
efficiency
the
non-standard
waffle
its
a
(1)
have
difference
equipment
1-2).
the
studies
structure
of
money.
and
testing
for
that
subassemblies
machined
resisting
as
isotropic
was
(3)
(2)
of
1o2),
an
orbiter.
thermal
used
concept.
major
this
model
obtained
structural
and
the
pattern
for
supplying
_'esistance,
isog_'id
Full-scale
(2)
piggyback
for
like
stiffening)
system,
at_aclled
acts
included
wings,
being
interiors.
NASA
integral
protection
the
it
by
now
(Reference
Workshop
funded
the
is
interstages
requirements
from
from
concept
since
(triangular
l_ad
and
results
study
thermal
structures,
verify
and
torques
loads
struction
local
shrouds,
The
external
could
and
isogrid
design.
stiffening
tanks
B design
booster,
fuselage
for
vehicle
structure
recent
shuttle
Delta
this
rib
design
that
a
intersections
structure
if
CR169
FORWARD
TANK
SKIRT
• ,- USE
This
1.8
CYLINDER
WELDED
WAFFLE
INTEGRAL
1.0
BONDED
HONEYCOMB
3.3
THRUSTSTRUCTURE
SKiN AND
(CONICAL)
STRINGER
4.9
AFT
SKIRT
SKIN
AND
";TRINGER
4.3
INTERSTAGE
SKIN
AND
STRINGER
14
The
pr,,sunts
handbook
flanged
structur4,s.
('(},.'_.rs
iso_ri(l
iso,_
The
basic
user
should
usinR
the
section.
a
inforr;_,;t:o:l
lu't'e!_,d
Sots1..
to design
is,)firid,
triangular
k,,'>- points
al)()ut
is(,uri({
ar,,
.,,t_)\vn
in
_tnd
is()_t'id
with
thv
inl(_rzll,,tioz'i
1- 1.
unflanRed
as
tL,kNI)t_()C)t,,.
stiffened
Table
BULKHEAD
Costs
TIIE
handhoot<
integral
on
OF
3.8
D MONOCOQUE
COMMON
Relative
STRINGER
WEt.DE
TANK
1-1. S-IVB
AND
DOMES
"-'-----C)
Figure
SKIN
ready
h_,in,,:
so
ct,.sien.tt,,d.
flaF=_vrl
-\11
()th,,r
iIlt'(_rzl,nli_,n
npt)li,,s
l(,
rid.
theory
f,,r
acquaint
handl)n()l<.
Th(-
unflan,.:,_.(',
both
basic
th,,
analysi._
Izill_s,,lt
B()th
th(,ory
ol
xxllh
tlllIl;tllt2_,(']
i-
,s()_'.'id
thi_
all(l
is
pr,'_,_'p.t,,rt
nnd
analysis
l'lnn_ed
su,'_11_ilrix_,d
r,'f(,renc(,o
1.0.003
at
its
iso,L:rid
th{,
cn([
()t
in
,":,_'cti(>n
2,
"]h¢'
a,_._tllilpti()llS
l)('i'Ol'("
a r_.
covered
t)y
th,,
._,cti(_i1
t_)
this
s(,r\_,
CR169
W
li(Itii_'
I ,_ N,_ii
i,l,ll,,l
llll
llt,,I,i!l,i!,l,i_
,,I | iIt_,l,ill,'lll
1,0,004
,dk
Table
i-
1
ISOGRID
0
A
lattice
of
triangh,
intersecting
ribs
forming
an
array
of
equilateral
s
Characteristics:
0
-
Isotropic
(no
directions
-
Poisson's
ratio
-
Efficient
in
of
inst,,bilitv
t) r weakness)
1/
conlp,,-cssi_n
and
bcndint,
Advantages:
-
Easily
analyzed
-
Can
-
Standard
be
optimized
for
wide
pattern
for
attachment
equipm,,nt
_nountine,
-
Readily
reinforced
-
Redundant
load
-
Less
In
use
and
on
for
of
loading
Inodt,_
witlmut
intcp.sities
a:'commodate
chan_t.)
conccntratt,
d
loads
anti
proorams,
'lilot'-l)clta
cutouts
paths
structural
dt, pth
two
.- t,,,_cc
n_aior
cxtunsix'cly
rang_,
invcsti_alud
and
tested
on
space
and
Skylal_,
shuttle
study
effort
Section
3 describt,
some
current
Section
4
space
vehicles,
and
chl,
lttlurt,
presents
structure,
Riven
s the
where
thv
analysis
worked
examples,
equations
and
_f
mvtheds
the
graphs,
accurately
size
isogrid
•
Spherical
cat)with
•
Cylinders
in
•
Cylinders
und_,r
vt, hiclt,
fo,"
siluatit,
ns
optiTr:i.<in,.:
art,
advantal2,'s
m,,Ihods
exist.
which
of
!or
tit,sign
nwthods
such
and
application:,
l'ypical
anti
raclt'ristics
the
"l h,,
gix',,n
to
"Ih,,
tzraphs
,,nal)lu
"l'h,,
structural
prt,
ssuru
slruclur,,.
r,,','(,rs(,
cotlq)rG.ssion.
torsion,_!
strucluru
,e,,,fi ( l_, tht,
1)_,ndin.tz
sh_.ar
'1.0.006
including
found
described
struclttr't,
nwthod
isogrid,
structure.
typical
at,,
of
of
for"
for
trs,,r
in
the,
u,-,,r
Ix, pus
each
nainimutal
analx'sis
is
llu,
ira at, totype
of
w,'ight
followed
art,
by
application
to
prest,
of
quickly
and
nt,,d
arc:
the
i_
Cylinders
under
In-plane
concentrated
load
in
an
infinite
In-plane
concentrated
load
at
the
edge
Cutout
isogrid
shear
Open
isogrid
cylinders
Open
and
structural
not
included.
To
comph,
weight
y.
a,
the
and
6 curves
to
butions
in
anti
o,a
information
hardware
and
in
covered
Power
brake
•
Creep
and
•
Compound
Referenct
s used
taken
to
it
becomes
pany
or
sheet
bending
have
not
been
is
a
analyzed
is
given
on:
(1)
to
the
and
minimum
compres,_ion
and
bending,
important
note
on
very
date
the
are
overall
and
(2)
off-
of
the
x,
use
of
node
flexibility
methods
sub-scale,
and
on
of
local
analysis.
full-scale
manufacturing
the
stress
distri-
Section
6 presents
Finally,
Section
testing.
techniques
developed
on
production
research
protarams
to
date.
to
new
of
data
manufacturing
7
The
are:
•
new
a
accuracy.
advanced
Machining
entire
axial
recommends
on
•
handbook
to
12
effect
model,
presents
4.
ensure
the
isogrid
This
of
plates
cones
subjected
Section
q d,..-cribes
topics
sheet
compression,
information
isogrid.
information
as
sections.,
Section
in
isogrid
such
cylinders
optimuna
pressure
w_.bs
skinned
types
te
for
external
reinforcement
Open
9
Other
uniform
age
forming
curvatures
in
is
set
sections
remove
tht,
text
are
listed.
up
to
allow
the
by
using
obsoh.te
available
NASA
forming
the
material
to
the
user
user
decinlal
page
innnediately
from
research
insert
pages
number's.
and
or
Care
to
add
development
test
or
should
be
information
in
his
as
com-
agency.
1.0.006
"
Section 2
BASIC
The
isogrid
rib-grids
mean
values
solid
continuous
It
shown
is
of
an
that
and
isotropic
layer,
in
if
key
the
properties
so
that
material
with
assumes
are
out",
the
averaging,
gridwork
appropriate
a uniaxial
constants
to
is
state
of
in the
of
taking
as
a
properties.
stress
those
or
considered
elastic
identical'to
rib-grid
and
analysis
bars,
an
isotropic
construction
is
the
material
is
character.
elements
of
the
individual
depend
upon
the
bar
to
act.
2.
1
HOOKE'S
isogrid
rib
The
gridwork
strain
are
the
orientation.
of
FOR
pattern
the
treated
constants
construction.
law
by
the
as
for
each
be
determined.
For
the
skin,
plane
upon
the
which
are
the
and
shown
stresses
For
they
strains
resultants
are
strains,
may
internal
stress
relations
composite
the
to
be
in
th."
bars,
these
dependent
stresses
upon
are
the
assumed
RIB-GRID
of
relations
that
The
These
ISOGRID
consists
assuming
elastic
determined
the
layers
normal
Hooke's
and
in
From
LAW
appropriate
construction.
in
the
co,nposite
skin.
composite
of
the
with
construction
the
orientation
combined,
material,
the
in
are
isotropic
the
"smearing
layered
composite
couples
ar_gles.
by
of
one
skin
viz.,
The
analyzed
stress.
ribs
The
grid
elastic
plane
When
the
sheet
smeared-out
in
are
THEORY
a network
are
the
of
equilateral
developed
individual
stress.
2.0.001
by
bars
(60
isolating
are
ir,
degree)
an
a state
tri-
elen_ent
of
uniaxial
of
P3
P2
×
X
/\
Element
By
means
of the
e
ei
one
obtaina
strain
cos
x
the
2
of Isogrid
transformation
O. + Y
x
xy
rela:ion
sin
law,
O cos
i
between
the
O. + e
x
Y
uniaxial
strains,
ex, ey
sin
bar
2
(2.1. I)
O.
x
strains,
e i,
and
the
x,
y
•
and _
grid coordinate
Rib Crid
xy
el
(Z. I. 2)
e2
4
0
I -q_
'3!
Note
known
that
(for
determined.
the
strain
example
t
i
from
strain
is
gage
invertable,
readings}
so that if (e l,
then
(e x,
Yxy'
ey)
e 2,
e3)
may
are
be
In fact,
1
ey
transformation
t'O i'''
0
2q'_-2
• 2
-1
Z
e3
(2. I,
_)
The
uniaxial
P.
bar
=
1
bee.
loads
are:
1
{2. I. 4}
i
--
Resolutes
of the
lengths,
grid
1,2,3
bar
a and
_tTa
loads
in the
give
the
x and
y directions
"smeared-out"
or
divided
mean
value
by
the
stresses
periodic
in the
element.
2PI
+ (P2 + P3 } cos 60 °
4Pl
+ P2
+ P3
(2. I. 5)
ff
X
(P_
+ P_)
y
sin
60 °
_
(P2
a
T
TXy
- P3 ) sin
60 °
--
(2.
P2
(2. 1.7)
'_a
(2. 1.4_
- P3
and
_2. 1.2),
2a
thrse
become,
{x}y
ih3 3]{}eyeX
v
xy
-r
yx
-
3
8
bE
h
-
_-
a,
the
triangle
12. I. 81
(2. I. 9)
xy
whe rt_
h
1.6)
--
yX
eq.
+ P3 }
2a
(P2
--
Using
2 _a
Ha
height.
1.0.005
By
comparing
isotropic
eq.
(2.
materials
1.8)
in
and
plane
(2.
1.9}
with
the
Hooke's
law
relation
for
stress,
(2.
(ry
1 - v2
v
y
E
ryx
xy
it
is
whe
evident
-
that
1.10)
(2.1.11)
2 (l+v}Xxy
eq.
(2.
1.8),
(2.
1.9}
are
a
special
case
of
(2.
1. 10),
(2.
1.
11)
(2.
1. 12)
"e
1
3
b
and
the
barred
modulus
2.
of
quantities
the
Z
ENTENSIONA
COMPOSITE
Many
con,-tL-uctions
cept
is
which
the
equivalent
Poisson's
BENDING
AND
STIFFNESS
FOR
SKIN
CONSTRUCTIONS
ratio
and
L AND
RIB-GRID
may
be
a two-dimensional
idealized
three-dimensional
thc
surface
loading
on
anti
stres_
couph's
elastic
approximation
the
Th(
as
is
obtainod
direction,
of
body
considL
by
red
to
integrating
plates
and
shells.
three-dimensional
This
Z
con-
elasticity,
by
a two-dimensional
be
resisted
by
the
stresses
and
surface.
stress
resultants
,laoments
in
the
Ov
Ny
i
Young's
gridwork.
replaces
thickness
indicate
My
y_
Y
Z
Nx
_i'iy
x
Mx
x
Mxv
$TREB
RESULTANTS
ON REFERENCE
gURFACE
ELEMENT
8TREU
COUPLE8
ON
PEFERENCE
EURFACE
ELEMENT
2.0.004
x
*l
These
and
are
y.
ence
computed
If the
small
surface
stress
per
unit
length
differences
at a distance
resultants
and
of the
in length
reference
of a surface
Z from
the
reference
may
be
written
cour)les
surface
parallel
surface
as
is
coordinates,
x
to the
refer-
neglected,
these
follows.
V
N
o"
x
X
N
N
Q
f
Y
x
T
xy
_y
T
I
Z
Qy
dz
(2.2.
1)
ZdZ
(2.2.
Z)
(2.2.
3)
ZX
"rzy
i
:
f'le:YJ
Z
whe re
N
By
= N
xy
use
of the
yx
and
xy
= M
Kirchhoff-Love
yx
assumption
of linear
strain,
f
(z)
x
M
IE X
_x
I
_xy
{^
(z)
Y
-Z
2×
xy
E
Xy
Y
L
\vhere
(_x,
_xy'
reference
surface
Hooke's
and
law
couples
expressed
Ey) are
reference
surface
changes
of curvature,
relation
for
and
fence
in the
refe
following
each
layer.
surface
The
strains
form.
2.0.0U
strains
together
and
with
relations
and
changes
(×x' 2Xxy' X_)
the appropriate
between
are
stress
results
of curvature
may
be
i:/[i0o0
:=lt
x]
Y
0
vD
(2.2.4)
Xy/
,.v[:o
l
,,_xyj= _MyxJ _
K is
the
extension_
K
"--L-ll
- v2
(2.2.5)
i sti'fness,
(2.2.6)
f
E (z)
dz
z
D is the
bending
stiffness
(2.2.7)
D
-
----_vl
1
J
E (z)
z2dz
!
z
and
the
reference
surface
J
E (z)
:
has
been
chosen
so that
(2.2.8)
zdz
0
Z
E(z)
of course
function
of the
The
integrals
known
as
Let
E o be
the
is the
thickness
appropriate
coordinate,
(2.2.6)
- (2.2.8)
method
of the
a constant
Young'a
reference
may
modulus
of ribs
or
skin
as
a
z.
be
evaluated
"transformed
geometrically
by
a device
section."
modulus.
(Z. 2.9)
K
t
Z
_tO.m
D
E/
Eo
_
,
zdz
I - i..2
0
--/
The
quantity
unit
section.
width
will
z2dz
Eo
E(z)/E
then
° may
now
be
thought
It is convenient
to take
Eo
be
the
will
1.00.
Only
_
rib
TRANSFORMED
of the
as
modulus
skin.
The
the
be
transformed.
]_
d12
d/2
hSOGRID
Let
t
:
b,
d
=: rib
skin thickness
w,
c
: flange width and depth
h
:
web
thickness
and
of the
-I_L
OF
depth
triangle height
2.0.007
11)
width"
_
SECTION
(2.2,
a "transformed
i®
bib
10)
of as
"
i®
(2.2.
WITH
FLANGE
skin
The
transformed
rib
Choose
an initial
The
normal
final
_
EE_
width
normal
is b/h.
The
coordinate,
g,
from
be
chosen
coordinate,
zdz
=
z,
will
trar_sformed
flange
the
midpoint
to satisfy
width
is
w/h.
of the
rib
web.
the
condition,
of the
transformed
of the
geometric
0
O
z
This
is
equivalent
to saying
that
z -- 0 is the
centroid
s e c ti on.
Define
the
following
d
-- 6t
c
= kt
{X
non-dimensional
parameters.
bd
th
--
Using
the
ties
of the
parallel
axis
transformed
a tabular
theorem,
section
appears
as
analysis
proper-
follows:
A.d.
I
A.
1
Part
_i
Q
t
t
Q
ta
0
t_
t
(1+5)
Aig!
Ai_iZ
t2
t3
-4 ( 1+5)2
T
(1+5)
0
_ (l+x)
1
o
2
1
12
(tz)
0
t2
3
--f_(l+_l
_ _(l+×_2
t3
_-_[l+ab2¢vk
i
2.0._
2]
"
...........
Then
_iAi_i
A
= S
Ai_
i2
4 _
io i
_A_
_
i
A
and
I are
sectio
respectively
the
area
and
nloment
of
inertia
of
the
transformed
-,(!
_>,I]2 [
.
A
=
__
t (1
* a
_ u/
[(1+6_ - ._(l +kl]
t__
2
1 + ,_ + t_
tZ 3(1+812
IZ
_[_l _
,)
+ _,p.(l_X.t
2 + 1 + o, _2
+ pk"
I + c* 4 bt
t
OF
I -
wher
t 3 _z
12 (l + _. +..'_'
e
(1
The
number
a nd
g.
+ a +-g)
of
independent
_(1._6)"_3t*(1+kl
+ 1 + c,t,
non-dimensional
v}_X.
paran..ters
2.0.{_
-
is
(1+6}-p
four:
a,
b,
k,
r ......
"!
1
From
(2.2.9)
eq.
and
(2.2.
10)
one
obtains
7
D
_8Eo
=
I
(2.2.1
t(
=
2)
EoA
since
v
The
is
:
1/3.
foregoing
analysis
assumes
also
1/3.
If this
condition
express
eq.
(2.2.4)
and
For
aluminum
Certain
terms
arising
from
the
eq.
(2.2.4)
and
(2.2.5).
Reference
the
not
satisfied,
in
the
Poisson's
it
simple
ratio
of
the
not
be
possible
will
form
shown,
skin
material
to
Reference
2-1.
= 1/3.
obtainable
twisting
from
rigidities
For
thin
the
of
rib,
the
these
foregoing
integration
bars
may
be
added
terms
are
negligible,
process
to
2- 1.
Z, 3
NON-DIMENSIONAL
ISOGRID
For
unflanged
the
v
not
and
on
(2.2.5)
materials
small
is
that
STIFFNESSES
isogrid,
preceding
k
=_t
= 0
in the
FOR
UNFLANGED
equations
developed
for
flanged
isogrid
pages.
1/2
:
For
_(_,a)
construction
a
=: b
:
: [3_(1+6) 2 + (1+_,) (l+,_a2)]
consisting
of skin
alone
(z. 3. 1)
(monocoque),
o,
(z. 3.2)
13
=
1
2.0.010
L
In
terms
of
a and
E
[3,
t
o
K
:
D
=
_
1-v
(l+a)
E
It
will
be
ot 3
(2.
L
12(1.
v2)
l+a
noted
that
E
(2.3.4)
t/1-v
2 and
E
o
bending
and
stiffnesses
[52/(1+a)
due
A
plot
of
_(a,b)
it
is
required
and
Suppose,
to
for
[52
1+o
of
=
the
t3/12(1-v
2) are
the
extensional
and
o
the
represent
nesses
3. 3)
skin
the
alone,
while
relative
increases
Figure
2-1.
the
non-dimensional
in extensional
and
factors
(l+a)
bending
stiff-
ribs.
is
shown
to
in
determine
example,
a and
the
This
graph
is
useful
when
[5 is
known
6.
required
D and
t are
known.
Ti_en,
C
where
C is
some
If this
relation
[3(a,
6)
graph
and
5
may
The
[5(a,
constant:
is
and
be
6)
optimum
2.4
MEMBRANE
For
many
are
negligible.
on
Solving
transparent
superimposed
read
graph
from
plotted
value.
on
the
for
[3,
paper
to
[5(a,
the
same
6)
graph,
acceptable
useful
for
off-optimum
[3, a
scale
as
the
values
of
a
off.
will
also
be
found
very
perturbation
construction.
STRESSES
conditions
the
The
changes
membrane
of
curvature
stresses
may
2.0.011
and
be
associated
determined
bending
by
sinl_,le
stresses
CR16W
4O
35
3O
25
15
10
Figure
2-1.
Plot
of/1(
a,
dl) Curves
2.0.012
equilibrium
conditions
classical
as
elasticity,
given.
Eq.
The
(2.2.4)
or
may
be known
Reference
problem
reduces
2=2.
now
is
to solve
from
plane
stress
Thus
N x,
Nxy,
for
the
skin
solutions
and
and
Ny
may
rib
stresses.
in
be
regarded
to,
(2.4.
Ny
[.1/3
1
e
while
Nxy
1
= _-K
K
= 9Et(l+a)
(2.4.2)
_xy
and
solving
for
the
(2.4.3)
strains,
1
ex
ey
1
These
1
-1/3
=1/3
1
}
(2.4.4)
Ny
N
8
×y
= "3 Et(l+a)
_xy
2.4.
Et(l+a)
Skin
Stresses
are
given
by
(2.4.5)
this
Hooke's
law
relation
for
the
{x) [ 3]{}ex
Oy
_y
=
_
1/3
1
skin.
(Z. 4.6)
ey
E
2(l+u) _xy
(2.4.7)
2.0.01;S
1)
By
use
of
(2.4.4)
and
(2.4.5)
one
obtains
the
skin
stresses,
O'y,
_x'
rxy"
(2.4.8)
_y
t(l+a)
N
1
•1"
-
If the
t ( 1 + _)
quantity
These
stresses
ultimate
buckled
One
the
equal
the
stresses
with
use
effective
skin,
is
determine
size
Young's
fixity,
in
ribs.
of
Triangle
Sorer
tests
y.
than
to
in
this
triangle,
the
the
been
vary
considerably
are
be
In
fixity
of
the
thickness,
the
the
conducted
to
determine
case
stiffness
in
of
the
case
of
under
In
the
the
skin
one
where
geometry
are
(2.4.
11)
may
of
of
the
panel.
skin
unbuckled
skin,
skin
panel.
the
stress
the
This
field
The
and
yield
effective
triangle.
ribs
the
with
compressive,
allowable
edge
10)
in
compared
effective
rib.
the
stresses
skin.
case
in
edge
stress
field
conservative
in
esti-
development.
depending
edge
fixity
values.
are
a little
more
complicated
that
the
upon
buckled
or
unbuckled
Stresses
tbe
fact
bat's
are
not
all
2.0.014
i,
allowable
may
this
buckling
and
have
stresses
the
the
the
they
determine
of
-"
xy
the
unbuckled
will
stresses
and
less
T
or
sizes
rib
,
concepts
upon
and
:
Y
(2.4.
N
tensile,
more
The
x
or
and
3
to
to
fixity
2.4.
due
0
edge
requirements
is
,
eff
buckled
modulus
depends
N
a portion
of
turn,
skin
Rib
t
width
as
the
N
is
treated
then,
If the
problem
to
defined,
are
constructions
upon
mates
be
stress.
is
depends
the
must
allowable
problem
the
_
the
material
is
x
skin,
may
t(l+a)
0"
If
consider
xy
tef f =
construction.
or
(2.4.9)
IXl
xy
than
oriented
the
in
the
skin
stresses.
coordinate
This
directions
From
eq.
(2.
1.3)
and
(2. 1.4),
T¸
0.1
One
13 2
b
°"3 -
133
b
eq.
(2.4.
_-
notes
0.1
0.2
If,
= Ee
2
0"
Using
P
1
_
b
=
E
4 (ex
-
+ ff-_ '_xy
E4 (ex_
'
1} and
that
:
x
if N x
_
+ 3 ey)
,¢xy
(2.4.2)
+ 3 ey)
these
become,
°l
:
1
3t (l+a)
(3Nx
a2
:
2
3t (1+a)
(Ny
a3
:
3t (l+a)
and
N
y
2
are
- N ,)
+
#3 N
(Ny
principal
(2.4.
12)
(2.4.
13)
(2.4.
14)
,J3 Nxy)
stress
xy)
resultants,
Nxy
0
t(l+=='=='=_),x -'_
:
0"3
2
3t(14
:
in addition,
Ny
::
Ny
0,
then
0.2
: 0.3
: 0
and
N
x
m
0.1
::
tel f
Note
in the
application
tion
of the
x axis
see
sketch
on
and
Page
of eq.
that
the,
(2.4.
12) that
the
1 bar
is oriented
2 and
_ bars
are
at
degrees
2. O. 019.
2.0.011S
±60
in the
to the
direc-
× axis,
For
example,
in
the
y
is
consider
a
circumferential
the
x
=
pR,
=
0
N
with
direction.
longitudinal
N
cylinder
In
internal
this
pressure
case,
xis
with
the
one
hoop
set
of
coordinate
ribs
and
coordinate.
xy
and,
=P_h
x
tef f
_2
03
-
2. 5
EQUIVALENT
Because
of
all
the
pR
3tef f
-
MONOCOQUE
the
isotropic
established
plates
and
shells,
E*
properties
AND
of
isotropic
solutions
References
2-2
to
t::"
the
construction,
from
extensively
it
is
possible
developed
to
theory
use
for
2-8.
in
In .many
cases,
stiffness.
In
these
other
are
expressed
in
cases,
however,
the
primitive
parameters.
For
equivalent
monocoque
thickness,
give
the
same
bending
and
such
extensional
terms
solutions
cases,
t*
and
of
it
is
the
bending
have
been
possible
to
Young's
modulus,
stiffnesses
as
2.0.016
and
(2.2.7)
extensional
reduced
determine
E":',
an
_hich
and
to
(Z.
will
2.6).
more
Thus,
E
K
=
A
o
E":"t_:_ :
1-v
1-v
E oI
A
and
in
Solving
2)
I are
expressions
the
t-':-" :
_=
E':-" -
E_,:,
Note
that
once
the
second
Thus
for
no
t ',_
=
t
E':¢ =
E
unflanged
(2. 5.2)
for
t::' and
E ::',
In
eq.
are
where
the
isogrid.
_
(Z.5.4)
the
first
factor
factor
(2.5.
3) and
(2.
5.4)
reproduce
the
requir(d
important
to
be
and
E
ribs.
to
inertia
(2.5.3)
no
only
of
t--_l+_
_ = I for
couples
moment
pertains
to
represents
the
the
skin
property
influence
of
the
and
rib
that
grid.
ribs,
c_ = 0 and
is
and
for
again,
Since
it
area
valid
non-dimensional
using
)
[5 arc
A
=
I+_
12(l-v
transformed
a and
(2.5.2)
2
l-v 2
(2. 5. 1) and
E':'
E ot 3
-
i2(l_v
(2. 5. l)
l-v
-
where
t
o z (i+_)
2 --
E ':-"t::'- 3
D
E
and
substituted
a word
bending
note
that
no__t to
stresses.
must
these
be
of
and
are
caution
extensional
related
Thus,
expressed
is
to
the
couples,
_.0.017
Since
t _:_and
D and
K,
resultants
and
stress.
into
t':" and
stiffnesses,
stress
equations
in terms
required.
of
stress
which
resultants_
and
E*
Use
E
Y
of t::= and
E::: for
geometrically
To
obtain
deflections
related
that
for
permissible
since
deflections
aro
it is
by
to strains.
a quantitative
experience
is also
idea
many
of the
magnitude
optimum
of t$ and
constructions,
one
i-,
by
E':%
has
found
approximately,
!,
a
(
Since
t"
this
=
1/3,
:
16
the
equivalent
:
t(l+3a)
implies
weight
thickness,
is given
(2. 5.61
an
equal
distribution
:
(-_)
(,4)
of rib
and
skin
material.
Thus,
Eeop
Kop t
t
E
= _- E °
E,(4)
= _
l_v 2
E
-_
(2.5.9)
t3
O
Dop t
:
12il-v
Z)
(2.5.
E
t3
O
Dop t
-
12(1..
19Z
v2)
1.0.0111
10)
Thus
the
extensional
by
a factor
of
by
the
stiffness
4/3,
addition
of
mate
r ia I.
Z. 6
SUMMARY
and
for
the
many
bendin_
optimized
stiffness
an
equal
weight
of
OF
BASIC
THEORY
material
constructions
is
increased
in
ribs
is
by
to
the
increased
a factor
of
original
19_._Z
skin
1
r
COORDINATES
RIB
ORIENTATIONS
t
q
!
t
=
m
2
I"
;3-a
b
GRID
Z. 6.
l
Non-dimensional
a
=
bd
th'
[3
--
[3a(l
GEOMETRY
Param¢'te,'s
-I-6) '_ 4 (1 _,_,= (]_,&4)]
I/Z
2.0.010
"="---
.........
2.6.2
Grid
Moduli
-b
E = _E,
1/3
7_ =
Z. 6. 3
Rigidities
Extensional,
K
:
-_
(z)
dz
:
9
_E
OA
:
K
Eot(l+a)
z
Bending,
D
:
_"
rE(z)
z2dz
-
= 9EoI
D
8
9
Z
Neutral
= 0
E(z)
=
centroid
Axis
of transformed
zdz
Z
Z. 6.4
Z.6.5
Equivalent
t::'
=
_
E:'.'
:
E
t ',_ and
A
oV
Composite
N
=
:
t -_
l+a
--
E
E",'
(l+a)
o--T-
Stress-Strain
K
1/3
Relations
e
2.0.m
area,
A.
T ....
l} [
Mx
My
Z. 6.6
= - D
×y
1/3
K
3 Xxy'
Nxy
-
Mxy
D
= _ (2Xxy)
Membrane
N
x
Skin
Stresses
x
t eff
N
O"
=
t'_.
Y
)
eff
N
= -._
tef f
Txy
teff
2.6.7
=
t(l+o)
Membrane
= A
Rib
= Transformed
area
Stresses
I Ny)
teff
02
=
Z
-----3tel f (Ny
+ x/_ N xY )
2
_3 ==_
2.6.8
Ec_uivalent
E
(Ny - _N
)
Weisht
Thickness
: t(l+3o)
2.0.021
Section
ISOGRID
Isogrid
is
a lattice
CHA
of
exhibits
so
arranged
properties,
hence
the
make
a complete
structure
used
as
lattice.
Because
of the
like
homogeneous
transformed
More
the
an
open
isotropic
to
property
an
equivalent
literature
detailed
to
finite
critical
areas
of
an
name
an
analyze
be
the
element
analysis
nodal
intersections
of
't.
can
into
and
the
of
needed
to
the
be
elements
that
a
skin
as
ratio
of
{see
shell
isogrid
examine
ribs
1/3,
mathematically
layer
behavior
is
to
1Doisson's
isogrid
substituted
bar
Intersecting
attached
material
gross
contiguous
of
"isogrid
effective
metals,
can
a:'ray
arrangement
homogeneous
expression
ADVANTAGES
whether
and
structural
transformed
available
forming
simplest
or
is
AND
the
stiffening
The
ribs
This
isotropic
most
RACTERISTICS
stiffening
equilateraltriangles.
3
Figure
equations
3-11.
in
structures.
local
stresses
in
bars.
CRIG_
-'_
_-
bl
"
0.51)
E - 10 11061
/
-
0.10
EOUIVALENT
SANDWICH
I$OGRID PLATE
Figure 3-1, Ilogrid Is Simple to Aoalyze
3.0.001
3-LAYER
_hP
Being
easy
to analyze,
shown
in Section
4.
the construction
Dasic
structure
intensities can be accomplished
of the effect of standardizing
technique
prove
has been
As
originally
of the
of the
here
was
to provide
readily
equipment
open
crew
adaptable
evident
occasionally
local
local
can
sections
4.5
and
4.6.
ior
a compressive
4.3
pounds
intention
the
structural
leaves
other
for
the
sub-
installation
a long-term
updated
with
newly
to the surface
sheet
(0.36 pounds
capability
is
As
an
is
Failure
are equivalent
per square
occurred
in weight
foot}.
The
weiffht
was
minimum
_fficiency
an
8-foot-diameter
to handle
pounds
per
a concentrated
within
large
weight
of the
of 2, 500
distributed
in unreinforced
to handle
with
load,
intensity
inherent
required
example
dissipating
This
a limit load capability of
at any nodal point.
the panels
be accomplished
load
was
to a
geom-
3-4.
of additional
Figure
and
and
without
advantages
for this construction
load
together
pounds.
has
refurbished
reinforcement
this
(see
periodically
aluminum
How
flat
scheme
nodes
The
walls
equipment.
loads.
20_ 000
This
in Figure
working
3-3,
of any
in static test although
skin
3-Z
in Figures
installation
applied normal
substantial
points,
equipment
requirement
is depicted
mounting
seen
permitting
the
of equipment
mounted
be
0.025
be
forms
space.
of the
that
While
can
module
experiment
removal
pattern.
continuous
internal
pattern
As
it was,
tank to
small -- about half
geometry
that
to fit the
at 750 pounds
this
Workshop
waffle
designed
250 pounds
Orbital
at the
as
advanced
(the
attached
exactly
will
is very
are
structure
design
in Section 4-13,
of standardized
and
to change.
It is
developed
program
a "pegboard"
rework.
base
shown
study
B shuttle design.
lattice
quarters
components
space
As
of load
a quick and accurate
standardization
in a hardware
;.sogrid
floors
etry
geometry.
range
as will be
applied to a large integrally stiffened propellant
applied
Skylab),
a wide
rapidly, allowing
in a recent phase
and
The
sizi._.gover
that the penalty of geometric
of o_-.epercent,
is also readily optimized
a hexagon
isogrid,
concentrated
is shown
provided
inch
ribs
and
cylinder
designed
required
only
tangential
24
by
in Sub-
inches
load
across
of
the
3- 5).
3.0.002
'
'
"'
L
CR 16"J
DAC
Figure
3-2.
O=stribut_ny
Co,_¢:_1_trated
Load
3.0.003
- 35333
DAC-
CR169
35334
@
F_gu,_: 3-3. Ent.pm,._
Attach{.'(I to N(}(t,'s
3.0.004
J
CRI_
• OPEN LATTICE
0.400
THICK
• WEIGHT
EQUIVALENT
0.025 SHEET
TO
0.070
0.22R
4.200
0.438
Figure
3-4.
Skylab
Floor
mentioned,
As
has
been
that
is,
it
loads.
can
and
either
with
fail-safe
struc_re.
and
then
propagated
obtain
or
lattice
present
fail-safe
lattice,
cutouts.
be
This
a complete
skin
has
the
same
reinforced
choice
offers
offer
exceptional
to
it,
in
either
lattice
t,_-nsile
loads
a
crack
Shear
lower
Thel'elocal
loads
design
flexibility
opportunities
to
or
than
systems.
is
rectangular
bending
handle
more
lattice
the
and
itself:
capabilities.
to
the
by
by
shear,
example,
flaw
structur,.'
compression,
for
joint.
at
is
locally
should
the
design
a
stiffening
If
the
lattice
tension,
can
assembled
around
with
a
redundancy
across
carried
case
such
rectangular
this
iso8rid
resist
from
Similarly,
skin
the
by
skin
discontinuities
available
Grid
effectively
Stiffened
fore,
Wall
DIA
and
made
lattice
system.
stiffening
systems,
it
weight
in
&0.(XB
isogrid,
or
in
redundant
separate
the
skin
skin
Since
should
Figure
be
3-6.
design
from
cannot
be
can
be
this
is
possible
the
not
to
the
T
_r
r_r_
3_
c_
O
--J
_w
c_
C_
C_
"r
u_
r_
LL
3.0.006
_q
CR169
REDUNDANT
STIFFENING
GRID
AND SKIN (ADHESIVE
BOND +
MECHANICAL
ATTACHMENT)
BOND
LINE
/
Figure 3-6. Fail-Safe Concept
The
rib
lattice,
the
skin,
in
effect
triangularly
where
torsional
to
all
surfaces,
kind
of
design
doors,
3-8
landing
show
In
of
a
typical
respect:
as
a
of
stiffening
are
torsionally
needed
can
closed
torque
box.
and
elimination
is
be
doors,
in
orthotropic
construction
In
structural
a purely
clear
taining
space
shell
to
door
jambs,
centroid
be
spaced
a torque
stiff.
box.
This
met
with
The
advantages
of
applied
cylinders,
occupies
moisture
structural
away
means
an
from
Therefore,
isogrid
of
that
situations
open
con-
inspectability,
entrapment
are
components
and
the
the
and
therefore
depth
system.
such
obvious.
as
speed
brakes.
Figures
bas
been
found
advantageous
the
same
of
is
optimumthis
frames.
the
for
This
form
sense
inside
isogrid
less
stiffening
are
of
panels
its
access
3-7
and
examples.
it
rectangular
with
surface
compression-loaded
another
and
second
can
gear
load
a
stiffness
instead
access
shear
forms
stiffened
struction
This
carrying
is
constant
depth
with
is
true
not
in
deeper
weight.
As
&0.0_/
the
they
an
even
but
are,
example,
capability
case
deeper
important,
The
compressive
waffle,
frames
in
when
more
than
many
the
a
both
so
th¢,
kinds
when
stringers.
designs
require
larger
the
recent
study
con-
the
r .................
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Figure
3-7.
Access
Door
Designs
i _
I
I
NEEDS
Figure
3-8.
Speed
i i
Breke$
3.0.000
INNER
SKIN
....
,
|
-.
{
:!
substituted
isogrid
for
permitting
reduction
The
of construction
conventional
construction
in fuselage
structural
depth
above
that
in a transport
airplane,
from
4 inches
to
obtained
by
simple
1.5
inches.
g
depth
zation
technique
structural
noted
was
described
in Subsection
space
efficiency
can
be
improved
if the
isogrid
shuttle
booster
study
where
per
inch
This
was
proved
in a space
tion
for
a 198-inch
radius
and
10, 000
was
2.25
inches.
Figures
3-9
and
construction
and
in this
optimize
case
techniques
sizing
must
to the
The
practical
a few
of the
a larger
be
final
employed.
As
so
of the
far
show
weight
ribs
the
depth
manufacturing
The
with
a unique
solution.
from
advantages
of isogrid
mentioned
are
to be
kind
of this
does
not
Iterative
of this
&O._
construcloading
analysis
refinements
time.
of
samples
specimen.
sure
flanged.
compressive
test
More
and
are
progresses
encountered.
optimi-
efficiency
a design
configuration,
applications
cases
step
Both
pounds
3-10
formability
in a single
4.2.
the
the
are
preliminary
in order.
above
uncovered
are
with
rf_
C
7
k
o_
30010
03
3.0,011
Section
4
ANALYTICAL
4.
I
SPHERICAL
CAP
WITH
TECHNIQUES
REVERSED
PRESSURE
P
The
spherical
cap
cut
off
by
a plane
The
load/in,
in
with
and
the
reversed
pressure
loaded
b']
sphere
is
consists
uniform
of
external
uniform
in
a portion
of
a
sphere
pressure.
all
directions
and
frequent}y
for
common
is
given
by
the
equation,
Nob
4.
1. 1
This
for
=
:
x
Typical
in
separating
ally,
::'It was
1964.
design
skirt
occurs
most
such
as
LOX
material
may
frequently
bulkheads
are
designed
for
reversed
compressive
however,
must
Situations
propellants,
and
the
pR
2
Design
situation
length
it
N
also
be
designed
this
design
(Reference
condition
2-9)
for
be
which
LH 2 tanks.
saved
tension.
pressure
stability
and
may
by
bulkheads
Considerable
such
designs.
For
some
loading
act
upon
the
vehicle
Gener-
procedures,
bulkhead
so
:::
under
the
compression
loading.
initiated
the
development
of
4.1.001
used
isogrid
that
in
Other
design
subjected
possibilities
to external
might
be
hydrcstatic
spherical
end
pressure
such
assumed
that
closures
as
in cylinders
vacuum
tanks
or
submersibles.
4. 1.2
Method
of Optimization
The
optimization
technique
all modes
of buckling
used
i. e. , general
ling are equally likely.
"one-horse
shay"
ious modes
of buckling
This
Instability_
Buckling
of a complete
optimization
occurs
and skin buck-
principle is popularly
It assumes,
when
in particular,
known
as the
that the var-
failure are uncoupled.
sphere
1
Ncr(1)
weight
instability, rib-crippling,
design principle.
General
minimum
may
be written
in the
form,
Reference
E tz
2-4,
(4. I. I)
=
3 (I - vz)
Since
and
eq.
(4.
1. 1) is
E-".' of (2. 5.3}
isogrid
"_n the
and
form
(2. 5.4)
may
of a stress
resultant,
be used
to transform
the
equivalent
{4. 1.1)
into
rean
formula.
l
Ncr(1)
=
_
t
(1 - vz)
R
E t2
(1 + a) 2
Et
Ncr(1)
E
2
(4. I. 2)
-3 (1 -
v 2)
4.1.002
i
This
equation
shows
the
typical
form
_hat
the
factor
gives
the
strength
the
nondimensional
first
increase
due
of
isogrid
of
to
the
the
equations
skin
addition
using
and
the
of
the
ribs.
is
customary
(4.
1.2),
a,
second
13 and
b,
factor
For
in
shows
typical
I
optimum
Since
designs,
_ = 16.
test
values
generally
fall
',knockdown
_ or
"correlation
Ncr(1)
With
=
(1
-
Et
2
to
eq.
t._ apply
Reference
a
?--8.
._
v 2)
(4.
co _
interpretation
eq.
Y,
it
2
--_
a proper
effects,
Et
=
_]3
theory,
factor,"
,_
Ncr(1)
below
(4.1.3)
of
may
also
be
c o to
account
used
for
for
the
spherical
reduction
caps
due
under
to
1. 3)
boundary
external
pressure.
Skin
Buckling
From
Reference
equal
biaxial
2-9,
loading
k
=
(7
lz
cr
k
=
the
buckling
stress
with
simply
supported
_2E
c
5.0
Thus
2
12
(1
=
-
equilateral
edges
is
given
triangle
b7
the
under
equation,
(4.
t
C
C
an
2
- .z/
k
in
5 0_
"_'--
2
=
4.62
v2)
4.1.003
1.41
From
(2.4.
eq.
ff
11) the
skin
p
R
=
Using
eq.
(4.
in terms
pressure
is:
(4. 1.5)
1.4),
R
Ncr(2)
of the
-2t (:i + o/
y
X
cr
stress
t
Pcr 2
-
= 4.62
Et
2
(1 +a)(a)
t2
(2)
N
= c 1 Et
cr
(1 + o)
(a. 1.6)
-_
where
4.62
c 1
Rib
= 3.47
Crippling
From
Reference
on three
2-4,
edge'
k
k
free
2
E
c
12 {1 - v 2)
:
cr
and
the
buckling
on the
stress
fourth
edge
Thus
2
0.456
C
12 (1
simply
supported
is,
,4
....
= 0.456
n
plate
2
(b)
_
C
k
in a long
2
:
0.422
v 2)
4.1.004
1 7'
From
eq.
(2.4.
12)
N
=
-
for
p
N
ff
Using
x
N
y
R
cr
-
2
Ncr(3
)
=0
xy
--':-
cr
O-
_
1
O"
2
=
g3
2
3t (1 +o}
-
Ncr(3)
(4.
I. 8)
(4.
I. 9)
(4.
I. I0)
(4. 1.7),
Ncr(3)
3.,,. ,[o
=
2]
2
=
C
Et
(I +o)_)
3
= -_(0.422)
c 2
=
0.634
optimum
Requirement
s
Collecting
formulas,
one
now
t2
co E _--_3
_
Ncr(1)
=
has
the
p
R
system
of equations,
cr2
t2
Ncr(2)
=
c 1 E t (1 +o)y
b
Ncr(3)
For
optimum
satisfied.
=
c 2 E t (1 + o)
Now
(4.
1. 1 2)
d2
I. 10) to (4. I. lZ)
these equations
are indeterminate,
are to be determined,
II)
2
(4.
requirements,
(4.1.
b, d, t, and h, while
4.I.0_
must
be simultaneously
in that four parameters
only three equations
are give.-.
As
a fourth
equation,
J
_
p is
the
speaking,
Its
for
ribs
_.ll
be
found
given
by
use
there
may
t,
required
pretation
for
a given
As
that
(4.
]. 13) will
yield
n__0.internal
pressure.
while
of eq.
use
phenomenon
implies
larger
of rib
(4. I. 13)
grid
a consequence,
an
optimum
two
classes.
less
than
the
pression-critical"
margins
effect
for
second
are
are
of requirements.
designs
are
called
"pressure-critical.
To
solve
(4. 1. 10)--
parameter,
N
-
E
cases,
growth
whose
the
pressures
burst
the
of flaws
optimum
pressure
the
those
For
example,
finite
skin
thickness,
The
physical
mean
ribs
for
increasing
which
designs
are
very
desirable
This
can
in cyclic
pressure
is
dominates.
skins.
may
result
pres-
increase
designs
inter-
thicker
deeper
the
all
than
designs!
exists
are
have
higher
t = 0.
occur
These
in excess
all
will
pressure.
of critical
state
so that
class
will
plastic
give
pressure
first
in the
counteracts
They
In these
loading
In the
optimum
pressure.
eq.
This
burst
of loading.
pressures
a,
designs.
prevention
class
sizes,
weight
region
some
will
material.
elastic
weight
higi_er
material.
ribs.
into
lower
of the
predictions.
Obviously:
is that
strength
since
conditions
to deeper
burst
in the
design
due
is
only
many
in skin
sure
holds
elastic
increase
designs
condition,
tensile
is conservative,
the
instability
is the
for
amount
until
1. 13)
Ftu
than
of eq.
in turn
sures
burst
loaded
of this
which
the
highly
fcr
be
(4.
and
conditions
more
be
pressure
eq.
burst
It will
is
consider
(4. I. 13)
burst
Strictly
use
may
pR
2t (I +a)
Ftu
where
one
in general
divides
all
whose
burst
called
pres-
" corn-
property
be
a very
that
important
loading.
In the
less
the
than
burst
These
designs
tile
non-dimensional
"
(4. 1. 13)
simultaneously,
introduce
N.
--
(4. I. 14)
4.1.11@6
from
(4.
-N
1. 11),
1. 13)
and
(4.
1. 141,
c I (_)2
=
From
(4.
(4.
(4. I. 12),
(4. I. 13
and
1. 1 5)
(4. I. 141,
(4. I. 16)
Multiplying
-_Z
=
(4.
1. 151
ClC2
h2d
and
_
()t.
=
1. 16/,
ClC2
th
]
2
=
Thus,
ClC 2
since
74-
o and
b are
positive,
o
(4.
Eq.
(4.
1.
buckling,
From
17)
and
eq.
(4.
satisfies
the
conditions
of
simultaneous
rib-crippling,
1. 17)
skin
burst.
1.
I0),
(4.
_" = Co N"
t _.__
1 +_
1. 13)
and
= Co
p
Ftu
(4.
1. 14),
( 1 + a) _
or
p
Co
(4.
( i. + o) 2
4.1.007
1. 181
Eq.
(4.
bur
1. 18)
satisfies
the
condition
of
simultaneous
general
instability
and
st.
If the
non-dimensional
x
loading
-()
= _
N
parameters
l 03
(4.
1. 19)
(4.
1 . 20)
(4.
I.
(4.
1.22)
_fc 1 c 2
w
2 N Ftu
-:
Y
CoP
are
defined,
then
eq.
(4.
1. 17)
and
(4.
1.
18)
become,
°(0,)
21)
b2
2
(I + a)
It
is
noteworthy
tions
of
the
this
the
x,
factor
are
and
consider
point
y domain
solution
the
right-hand
sides
Boundary
conditions
geometry.
correlation
at
that
will
not
the
instead
thus
have
involved.
of
solution
that
this
equivalent
Usln_
eq.
(4.
:
pRIl
2 Ftu
}-
l<
weight
1. 13)
? Ftu
thickness,
this
_ 3a)
(1 * o)
t,
are
the
plate
elements
reason
it
a mapping
a
simultaneous
is
independent
is,
becomes,
,
_ ]l + _
4.1.0{m
L
equations
as
i
The
these
for
For
attempting
a validity
of
i.e.,
of
is
the
c O,
funcand
convenient
o,
to
_, domain
solution.
of
pure
Tile
c 1,
and
stop
into
mapping
c 2.
One
now
has
parameters,
By
varying
curve
may
the
complete
p/
Ftu
solution
and
p/
Ftu
for
be
constructed.
in
terms
of
the
nondimensional
loading
p cr / E.
a given
value
of
p cr /
the
E,
nondimensional
weight
I
I
FOR
I
6
v
P
Ftu
MINIMUM
Define
the
pressure
computed
and
for
associated
is
given
on
log-log
in
If only
Such
optimum
pressures,
4.
graph
paper.
is
information
design
burst
i.e.,
if the
_,
of
Figure
On
5,
the
of
master
curve
(Po/Ftu)
may
be
constructed.
desired
and
if
P/Ftu
<
all
that
is
if
the
complete
greater
than
p,
is
is
pressure
critical,
on
4.
Figure
(Po/Ftu)
a
seen,
hand,
and
(per/E),
be
usually
CURVE
If (_'min/R),
may
pressure,
given
values
Po"
As
other
design
as
1-1.
is
graph
{_min/R}
sequence
(t-rain/R)
studies.
y;
a
for
WEIGHT
these
Po/Ftu
this
required
in
plot
graph
is
the
minimum
will
be
necessary
obtain
t.
This
graph
read
,_f'f the
to
of
as
is
preliminary
geometry
it
1-2
curves
are
ffrnin/R).
The
graph
straight
lines
sufficient.
design
required,
weight
is
9.2 if
the
pressure,
to
done
weight
use
in
the
the
Po,
x,
following
steps:
A.
Compute
a nd
x ,,nd
y
and
from
the
,'.
4.1,000
correq-onding
a
_11_l'_'_'r_'-_'_'_
_
__
v
o1_
.0
il
ii
ii
0
o
4.1.010
Y
1
o 1
0 2
() 25
0 3
i_ll_l_
2O
25
30
l
-¸ -
40
50
60
70
14
6
....
........ ):i_i
•
÷
25
30
40
50
60
70
CR1G';
20
25
40
30
50
60
70
80
9O
IO0
SUMMARY
30
TTI_-I
_.J,
I-:
OF DESIGN
EQUATIONS
FOR SPHERICAL
CAP
25
REF PAGE
_,t1_IIIilli',|
T ;[
_L_'lll!l
"'t:HI
i
2O
[
4.1.011
_
[ I!IITII
ii,il
i+H_H{
[llll
['.liil
',',',',I ![I
;!I,'I_LL
,lllll
"---"
_'_'-"
=
0.260
C1
=
3.47
4.1 .O03
i
_t!
15
C2
@tl
_.-},_
Co
4.1.004
0.634
REF
.....
EO
lll|lli
Illlili
(4.1.14)
10
(103 )
(4.1.19)
9
1.482
8
(4.1.20)
7
PR
=
6
2Ftu
-' NN
(1+
a )
(4.1.13)
14.1.26)
d
=
St
b
=
_-
d
h
=
_
t
(4.1.27)
(4.1.25)
AS A CHECK,
i_ii:
k
_"
=
bd
th
=
t (1 * 3a)
2
0.40
a
[lill
II1[I
*[I
il
II
II
1
l,lt
tll.,_[[
20
;1
25
311
4O
50
6O
70
80
90
luO
F,gure
4.1.011
4.1-2.
Design
of t togr id Spherical
Bulkheads
B.
t may
now
weight
be computed
pressure,
t - 2 Ftu
C.
Knowing
D.
The
t,
rib
from
the
burst
condition
or
from
the
Po"
pR
(1 +_)
the
(4. 1.24)
triangle
depth,
d,
height,
is
given
h may
by t and
be
computed
from
eq.
The
rib
width,
b,
is
computed
from
(4.
1. 15).
(4.
1.25)
(4.
1.26)
6,
d = 6t
E.
minimum
eq.
(4.
1. 16).
(4. 1.27)
As
a check
on the
computed
value
of a read
Finally,
the
from
Reference
2-8
For
heavily
domes.
off
value
values,
the
x, y: a,
to be
used
as
ratio,
bd/th,
should
agree
with
the
6 graph.
for
a function
stiffened
the
the
correlation
of the
ratio,
domes
the
result
x, y; o,
6 curve
factor,
t*/R
for
¥,
may
lightly
be taken
stiffened
of test
in Reference
in Figure
4. 1-2.
1-1,
gives,
¥ = O. 425
This
c o = 0.612
Y = 0.26___._0
is the
used
value
for
the
4.1.012
:1::.,_.._
..........
"--"--_
.... %.-
,,., ._=:_2°i
4. 1.3
Worked
Worked
Examples
Example
1
Pcr
= 21 psi
Pbur
st = 60 psi
R
= 96 in.
E
= 11.6
(106)
Ftu
= 78.5
ksi
Pc____r
E
21
6
= 1-T_.6 (10-)
From
1.81
(lO 6 )
mi.____n
R
=
0.000805
87.9
psi
60.0
and
the
(tmin/R)
=
graph,
103
=
Po
The
p si
design
rain
If this
is
is
-Pcr
N = _
x
=
"
12
= 1.1Z
(78.5)
=
compression-critical
= 0.000805
is all
geometry
1
that
x 96
: 0.0772
in.
the
analysis
is desired,
is completed.
required,
IFtu/
_Po
-N (1 o 3 )
1.482
:
/
1.81
(10 -6 )
1. 12 (10 -3 )
1. 617
= _
:
:
1.617
1.09
4.1.013
................................
psi.
(10 -3 )
value
is valid.
However,
if the
_..._#__{__,_1.617(lO
: o. _30k-Co/= o. 130
=II.I
From
graph,
a
= 0.275
8 = 16
Po R
I. 12
= 2 Ftu
(l+a)
d
=
16
b
=
bt
=
=
d = %_--
(0.675)
2h
a = _'--
As
a
This
2 (1.95)
I. 732
0.675
0. 04?-2 in.
in.
0.634
=
t = 4
= 46. 3 (0.0422)
=
2 (1.275)
(0.0422)
0.634
= 0.0505
=
(10 -3) (96)
(0.675)
0.0341
3.47
1.617
(103)
=
in.
1,95
= 2.25
_0. 0422)
in.
check,
bd
th
= 0.0341
0.0422
is
very
(0.675)
(1.95)
close
to
the
= 0.280
graph
value,
a=
0. Z75.
value,
train.
= t (I + __a) = o. 0422 [I _ 3 (0. 280)]
= O. 0777
.,
'('_
in,
4.1.014
As
a check
on
the
"train
Use
of the
burst
pressure,
87.9
psi,
would
have
Example
2
60 psi,
resulted
instead
in thinner
of the
skin,
optimum
smaller
pressure,
triangles,
and
heavier
weight.
Worked
Pcr
= 8 psi,
Pbur st :
= 1 Z0 in.
R
11 (106 ) psi,
E
= 76
Ftu
ksi,
8
=--(In11
Per
E
From
75 psi,
6
)
0.726
(10 -6 )
(train/R)
:
:
graph,
10
Po
= 0.644
This
design
than
the
6 curves
is
= 0.644
(76)
= 48.9
to obtain
(10)
In this
since
the
case
it will
_.
N = Pcr
--_-- (_)=
x
< 75 psi.
l)ressure-critical
burs__.__tpressure.
= 0.735
psi,
0.000466
0.726
(10 -6)
[76 75
(103)]
-3
(103 )
O. 735
=-1.482"
= 1.'T-./////_ = 0.496
4.1.016
minimum
be
weight
necessary
pressure
to use
is
the
a,
les__.__ss
3511o3,[76
i,o3,1
y o13op o.70.,3o
= 5.73
]_rom
graph,
t
=
a = 0.066,
in.
d
= 6t = 11.6
b
= _/_d
N
(0.0555)
= 0.644
in.
(10"4)
= _ .7,'350.634(0.644)
= 0.0341
As
11.6
pR
_
75 (120)
(10 "3)
2 Ftu (I + a)
2 (76) (I.066)
= 0. 555
a
6=
(0.644)
= O. Z19
N
in.
0.735
(103)
in.
= 68.8
(0.0555)
= 3.82
2h
=_,=
2. (3.82)
_
= 4.41
(0.0555)
in.
a check,
bd
th
- 0,0219
0.0555
10.644)
3.82)
This is close to the graph
= 0.0666
value,
o = 0. 066.
The t"is given by,
_" = t (1 + 3a)
= 0. 0666
= O. 0555
[1 + 3 (0. 0666)]
in.
4.1.01e
If the
design
t-rain.
It
will
the
had
been
compression-critical,
= 0.000466
be
found,
optimum
(120)
for
many
than
excessive
weight
penalties.
Spherical
Grid
The
analysis
detail
in
The
tables
developed
here
for
convenience.
Layout
The
geometric
by
had,
theT/R
means
that
curve
is
fairly
flat
the
skins
may
be
sizes
result
that
larger
grid
beyond
made
without
the
tables
used
to
lay
out
the
grid
is
described
in
1-1.
there
and
the
description
of
on
spherical
their
usage
is
repeated
Isogrid
layout
Consider
to
Reference
that
so
have
Layout
leading
of
plished
designs,
optimum
would
in.
This
thicker
1.4
= 0.0560
Po/FtuPressure.
somewhat
4.
one
of
the
the
following
an
triangular
gridwork
has
surface
is
accom-
routine.
icosahedroninscribed
solid
the
20
equilateral
in
the
triangular
figure.
N
/
4.1.017
spherical
faces
surface.
and
is
This
shown
regular
in
the
A typical
face
A view
of the
figure
as
is labeled
NAB,
equilateral
triangular
seen
from
the
where
apex,
N is the
whose
apex
base
(North
plane
is
Pole)
NAB
of the
sphere.
is
shown
in the
E,
and
N.
N
B
E
The
midpoint
further
of the
subdivides
arcs
the
and
three
congruent
arc
lengths,
from
N_or
C,
From
each
vertex,
nected
by
basic
AB,
b,
are
etc.
and
BN,
triangle
isosceles
a and
great
NA,
are
into
spherica!
labeled
central
b.1 for
points
along
as
C,
equilateral
NCD,
subdivided
a i,
corresponding
one
triangles
symmetrically
are
designated
This
triangle
AEC,
from
D.
and
each
CED,
BDE.
vertex,
The
i. e_
n subdivisions.
the
adjacent
arcs
are
cor.-
of the
elementary
circles.
D
1
2
C
3
1
E
The
arcs
will
intersect
in points
which
define
triangles.
4.1.018
the
vertices
The
subdivision
and
are
a i, b.1 have
cumulatively
shown,
been
computed
added
for
unit
radius,
for
in Table
4-1,
from
a vertex,
tabular
values
by R.
for
Spherical
Cap
lay out.
For
spheres
4. 1. 5
of radius
Summary
R,
of Design
multiply
Equations
Ref.
Page
4. I. OIZ
C o = O. 260
4. !. 004
C 1 = 3.47
4. 1. 005
C z = O. 634
(103)
X
Ref.
Eq.
(4.
1. 14)
(4. 1. 19)
I. 482
N
Y
(4. 1.20)
0.130
=
\p
pR
2 Ftu
!
(4. 1.13)
(I + a)
R ef.
d
= 6t
b
=__d
Eq.
(4. 1.26)
(4. I. 27)
(4. 1. 25)
As
a check,
}-
=t
(1 + 30)
4.1.019
n = 5 to
g0,
to facilitate
T
!
.a.I
..0
_d
[_
¢;
6
¢;
c;l ¢; ¢;
I
1
--
!
I
,!
I
4.1.m
•
p
_J_
o
•
!
)
o
!
,,-4
,.o
I
4.1.0_1
4.2
CYLINDERS
IN
COMPRESSION,
The
compression
on
the
moment,
N x,
lvi,
the
cylinder
a
resultant
ends
of
the
cylinder.
given
by
the
equation,
the
cylinder,
is
N
F
= _
*
2 _I_
M
_
cos
nR 2
maximum
has
two
in
x
The
at
BENDING
value
of
N
occurs
force,
The
for
cb=
in
design
F,
internal
and
resultant
axial
load/in.,
0°
X
N
4.2.
x
1
F
= -2-R.
(max)
Typical
A very
common
cylindrical
in
i_to
the
4.2.
Z
The
optimization
failure
and
shape
cylindrical
of
vie,',
modes
occurs
living
and
com_artmeHts
are
since
the
is
isogrid
to
especially
may
of
of
subjected
configuration
cylindrical
Method
Situations
application
payload
point
2
-R
Design
tankage,
The
M
+
be
fuselages,
space
vehicles
maneuver
and
attractive
machined
interstages,
in
the
that
thrust
arc
loading.
from
the
fabrication
fiat
and
then
for
simultaneous
rib
crippling.
shape.
of
in
Optimization
assumes
that
general
instability,
minimum
weight
skin
4.2.001
buckling,
occurs
and
formed
General
Instability
In
Reference
2-1,
due
to
bending
N
(1)
it
may
is
shown
that
be
written
in
theoretical
the
values
for
general
instability
form,
2
-
1
cr
This
In
_3
(1
theoretical
the
case
length
Reference
Fl(igge
If,
however,
combined
least
to
r
or
dependent
2-1
for
is
independent
of
the
described
on
an
R/t
was
2.
1)
by
both
::_ ratio
first
R/t;::
of
by
length
of
theoretical
the
cylinder.
critical
load
a looped
"festoon
curve,
and
and
been
85.5,
noted
the
L/R
has
a typical
R.V.
is
" Figure
plotted
value
for
isogrid.
in
1914
and
Southwell
highly
4.2-1.
from
The
later
by
1932.
bendin_
pressure
is
axial
compression
and
present
25
percent
of
the
value
given
less
than
10,
then
the
combined
then
the
axial
curve,
(4.2.
F
is
internal
interaction
eq.
is
in
(4.
R
compression,
and
dependence
W.
v2 )
of uniform
curve
length
-
formula
dependent
This
Et
1) (see
Figure
by
or
eq.
loads
component
4.2-2).
.._
I
4.2.002
if
the
loading
consists
where
the
bending
component
(4.2.
1) and
are
may
on
the
also
the
L/R
linear
be
ratio
of
is
is
portion
expressed
equal
of
by
the
at
!!
o.
8
u
m
I
0
0
m
I
;=
_o
0
r_
#
.J
\!
m
C
C
o
too.
E
m
0
m
/
W
J_
l'-
e-
C
{I
_.o
0
I_NIe
N
4.Z.003
?
|
C R IC'-"
I
NCL
Et "2
=_
R
= (CLASSICAL
R
t*
VALUE)
= 85.5
0.6
Nb
NCL
0.4
0.2
-_ =6.04
L
_
0.2
0.4
0.6
0_
- 4.2.
1.0
1.2
1.4
NCL
Figure
4.2-2.
The
axial
load/inch,
N
F
2_R
The
Interaction
a
bending
Curve
Showing
N a,
the
is
Effect
of Length
given
by,
(4. 2. 2)
load/inch,
N b,
is
given
by,
M
Nb=
may
3)
R_
Assuming
N
(4.2.
the
be
v_lidity
given
condition
by eq.
for
(4. 2. 1),
uniform
the
combined
compression
loading
to be
condition
met
so that
is,
a
2
E':,'t ':_
1
Na
+Nb=N
cr
(l)=_
3(1-v
2)
R
4.2.004
(4. 2.4)
The
theoretical
values
of eq.
(4. 2.4)
or
"knockdown"
factor,
¥,
( ¥ < 1.00)
allowable
compressive
geometry,
material
the
ideal
the
test
values
Using
the
N
values
of E ::_and
+ Nb = N
(1)
7 is
one
moderate
or
is
may
heavy
recommended
optimization.
Ncr
t _:-"from
use
theoretical
values
accounts
conditions
These
(2.5.
for
of test
deviations
to
deviation
of
specimen
always
from
reduce
3) and
(2. 5.4),
(4. 2. 5)
2)
correlation
a
a "correlation"
Et2 _
---_--
_ 3(1-v
7 from
factor.
Reierence
stiffening
a value
by Reference
2-11.
For
2-6
very
as
lightly
a functionof
stiffened
t'::/R.
For
of
This
is the
value
assumed
in the
Thus,
(1)
2
t"
= c o E_-
(4. 2.6)
where
co
eq.
_/
an appropriate
cylinders,
theory.
by
predictions.
=
cr
multiplied
factor
boundary
in the
theoretical
be
to convert
This
and
assumed
below
a
where
loads/inch.
properties
condition
must
= 0.612
(_)
=
0.397
Skin
Buckling
The
critical
a
stress
-
for
skin
cr
....r'2
E _a
12 (1-v 2i
the
formulae
Using
written
buct::':ng
is given
in the
form,
Reference
2-9,
1
(4.2.7)
'
for
isogrid
the
critical
skin
buckling
load/inch
may
be
as,
t2
(4. 2.8)
Ncr
An
(2} = Cl Et
appropriate
Reference
c
value
for
c 1 established
by
test
on optimum
stresses
will
for
principal
structure,
2- 1 2,
= 10.2
1
Rib
Crippling
Since
the
N
= 0,
and
al
= teff_ /Nx
xy
(1 + a)
maximum
eq.
(2.4.
x,
12)
occur
stress
conditions,
become,
1 )
- _ Ny
2N
(4.2.9)
2, 3
These
equations
1 ribs
are
Na
and
oriented
+Nb=N
if internal
N
y
show
that
axially,
ifx
is
then
chosen
as
since
x
pressure
is present,
then
=.pr
2
4.2.IXN!
the
axial
direction,
i.e.,
the
thus
(4. 2. I0)
_1
In
this
1 rib.
1 rib
in
tel f
case,
the
It
apparent,
is
the
:_
z 1
internalpressure
that
hoop
direction.
tef f
_
in
this
is
contributing
case,
it
an
would
be
additive
better
load
to
to
orient
the
the
Then,
+-_(N
a
_ N b)
]
2
2, 3
This
is
2,
3 ribs
In
the
(N
3 telf
a much
are
+
a
better
less
optimization,
(4
2. 1 I)
and
the
Nb)
arrangement
highly
it
since
the
1 rib
is
now
in
tension
stressed.
will
be
assumed
that
the
rib
stresses
are
given
by
the
relation,
Na
cr
-
+ Nb
t
(4.
2.
12)
(4.
2.
1 3)
eff
thus
b2
Ncr
(3)
= c2Et
(1
Ncr
(3) = Ncr
(I)
+ a)
j
where
4.2.007
From
Reference
cz
an
appropriate
is very
close
value
for
c 2 is
: 0.616
This
coefficient
tions
at the
In going
would
2-12,
attached
edges
to the
of the
value
for
unflanged
isogrid
ribs
to flanged
become
supported
by the
flange
and
by
Collecting
formula,
of
one
boundary
condi-
the
isogrid
value
ribs,
the
edge
for
c z may
"free"
be
(2)
= c
Et
cr
1
Ncr(3
) = c 2 Et
expected
10.
has
the
system
of equations,
tz
) = c O E_--f5
Ncr(1
N
a factor
support
plate.
from
to improve
simple
(4. 2. 1_)
(1 + o)
t2
(4. 2. 15)
h-_
b2
(1 + _) -7
d
(4. Z. 16)
where
Ncr(ll.
Eq.
= N cr (Z) : Nor(3)_ _ = N a + N b : N cr
(4. 2. 14}to
(4.2.
16)
are
formally
of the
spherical
cap
and
the
optimization
same
way.
To
eq.
pressure
Ftu
(4.2.
14) to(4.
may
=
be
Z. 16) append
regarded
as
identical
procedure
the
a "free
to the
equations
proceeds
"burst"
condition
parameter.
"
pR
t (1 + o)
where
for
buckling
in exactly
the
the
burst
(4.2.
4.2.01J
17}
Define the non-dimensional
loading parameter,
N.
cr
(4. 2.18)
From
eq.
(4.2.
(4.2.17)
and
(4.2.
18),
2
-N=c
From
15),
(4. 2.19)
1
eq.
(4.2.
16),
(4.2.
17)
and
(4. 2. 18),
2
(4.2.20)
Multiplying
_2=
and
eq.
o and
_Eq.
From
and
eq.
-
N=
5 are
and
(bd
th
(4.2.20),
)
2
t
(d)
4
= ClC2
a2
6--4
positive,
1c2
(4. 2. 20)
buckling
19)
b2t 2
h2d 2 _ ClC2
ClC2
since
(4.2.
(4. 2.21)
satisfies
the
condition
of simultaneous
burst.
(4.2.
14),
±_L
c o R l+a
:
(4.2.
17)
and
Co .2._
Ftu
( 1 +ai 2-
(4.2.
18),
rib
crippling,
skin
or
x p /
Eq.
(4. 2.22)
burst.
Define
the
(4.2.2z)
il+o)2
satisfies
the
condition
non-dimensional
of
simultaneous
general
instability
and
loading,
I
N
x =
_]c I
(10 3)
c2
(4.2.23)
i
N Ftu
(4.2.24)
cOP
then
eq.
(4. 2.21)
x -
and
(4.2.22)
become,
(103)
62
(4. 2.2 5)
(1 +,_)2
The
right
of the
The
hand
(4.2.26)
side
of eq.
spherical
cap
and
equivalent
weight
(4. 2.25)
the
same
thickness,
and
x,y;
t',
(4. 2.26)
a,
is
t = t (1 + 3a)
Using
eq.
[-=loR
Ftu
(4. 2. 16) this
becomes,
1 + 30
1 + a
4.2.010
are
6 mapping
identical
graphs
to equations
may
be
used.
or
As
±=.._
3..........._
1+
R
I + a
Ftu
in the
function
case
(4.2.27)
of the
sphere,
the
for
a given
value
of P/Ftu
quantity
t--/R may
be
minimized
as
a
of N C r/ER
R
u
I
I
I
MINIMUM
WEIGHT
CURVE
I
P
6
L
v
As
before
A.
this
will
all
design
into
two
classes.
For
p bur
the
B.
divide
st
Po
Ftu
< Ftu
minimum
weight
is given
by Po and
"compression-critical.
,,
of Po will
designs
and
additional
burst
effects
due
to
cyclic
Use
loading
safety
the
factors
on flaws
are
construction
thus
lighter
weight
so that
crack
propagation
less
severe.
For
Ft u
Ftu
the burst pressure
must
"p1 essure-critical.
"
be used.
4,2.011
These
is
give
Pbur st >
j,
Ft u
designs
are
If
a family
of
different
(tL_in/R)
N
ER)
( cr /
This
is
shown
The
complete
A.
_n
a
geometry
x and
a and
5.
t may
now
mum
weight
associated
master
Figure
Compute
B.
and
be
(po/Ftu)
values
non-dinlensional
4.
2-3.
is
determined
y
and
computed
the
Figure
4.
the
burst
from
pressure,
curve
by
from
are
may
computed
be
constructed.
following
procedure:
2-4
off
read
the
condition
for
or
corresponding
from
the
mini-
Po'
(4. Z. Z8)
t =
p is
the
larger
t,
the
triangle
where
Flu
pR
(1 +a)
C.
Knowing
value
_f
Pburst
height,
h,
or
Po"
may
be
computed
from
(4.
2. 19)
(4. 2.29)
D.
The
rib
depth,
d=
6t
rib
width,
d,
is
given
from
t and
5.
(4. 2.30)
The
b,
is
computed
from
eq.
(4.
2.20).
m.
As
a check
read
off
4.
2.3
on
the
a,
5
the
Worked
Worked
computed
values,
bO/th
should
agree
graph.
Examples
Example
1
R
:
48.0
in.
E
:
11.0
(106)
F
:
._O0 k
M
:
8000
psi
Pburst
Ftu
k in.
4.2.012
: 55 psi
= 67.0
ksi.
with
the
value
of
a
+,1
@
t ¸
o
d
0£
" !:
!.r
::,
- 1
.
l
{77
|
--
4¸
ii
T:G
::i
t
:_ .;.
':;
:il-_il
:.
:
o
u
..
,
•
:i]_-
,
:
:
i !
0
.4 .....
+.
......
+
I.i. ":
: ',;L;.
-
;i
-- -4 j
_;
._.!_
_--
_.:
"!
:"'i_'_
'._T
.... :.i
"_-_-1
:.;:
_.._.,
:;"
_
/
.
,. .....
-t
--"
1' i; !It;
_ _ _] *_
"---
• !
I
il:
X.L
I:
°,
i!
t_:
,!
8
O
i.i. : i
_
i.9:t
.:I:':
E
.;:.:';::
: : :
'
;,::
.....
,:i
;iZil
:iiii__
!
';!!_i
;..:: ....
,_
i[il'.[_;
--!--; !;-_,;*tt4+;t÷{
:.i!, ::_Ji;Hli:tll,
i::1
,,,
'
i ii!m_
l,.
;;:
L_
I i !
Pill "'
0
N, '
'_::i!! iul
ki_
,i _'. ,i,t N
cO
i|½
E
"r_
.....
'
,i
. _.,,,i
_+tt#+t
;LIT ,...1,..,I ....
L..
7_.!:;.'llM!lliill'.J
!_;i_...ii;
ii it i] i,t i iI
i:-.
l_
,_,,'.Lliiiir=iiil!l: i I
_: :i!t_;'41i!i!lli! i I
r,
_i"
_]:;IJiiitltll_tl
E
i
I_i+,
i !
. _,
I_
;4.1.:
'.'_
e-
::, i!!
i_ !_:!l!::L:li!_il:
!i!t
&
0
rj
,.i_
o
4.2.013
O
25
OJ
04
;;
0
30
.......
05
040
06
01
08
U9
::::
30
15
lO
0.05
9
8
7
0
(_
(.,3
!!_!!iil
iii!:'i:
_) 0
1
I
?
2_
4
3
b
4O
6
bU
bO
_-i-__l_ i
14
........
....
f
!I :_ :_:
iii!!!!i!i:ili_iii: ....
....
4
0.20
4
70
80
90
100
CR16'I
25
3U
70
4O
90
80
100
__...............
SUMMARY
OF DESIGN EQUATIONS
COMPRESSION
AND BENDING
30
FOR
CYLINDER
UNDER
AXIAL
REFPAGE
7!!',
!liii!!,.!!!
P.^
"._
;!ii',;_ili;
'1
0.397
4.2,005
20
i!TI!;!I
C1
=
10.2
4.2.00G
C2
=
0.616
4.2.008
REF EQ
i llliiilli!lllilI
(;II[!ilililIliii
1,5
_,
i !II!iiI_IIII!I![
14.2.171
, ;liiilllllllllll
l ilti!
lillllltll
IiiiIllJlilllll
I tli!
liil_
(103)
fi1111111111.
i illlllltlllll_ilJ
(4.2.22)
2.505
10
Y
0.397
;
i
i
[
i
ii!i
[ii
i
_111
:il'
111
iilll
PR
8
t
(4.2.27)
=
Ftu (1 +li)
I 'i;:i,>-_
III;III
l_llllilil!llllli
I_iltl,!lllll[l
7
d
lililill
I
;::_!iii
i ;'II_!,,I
: "iii;:
illiil.
(4.2.291
81
I
!i!-ii}i
6
ii_i,!
_,
iiTI:i;',
b
=
. n_n_n
(4.2.19)
J
(4.2.28)
;il
i
I
i
II
!t..,.
AS A CHECK,
f ,t-fl!._,
bd
i
01
T
=
th
t (1+3101)
"I-I I I_! i,L ! I i ; ! ! I_ ! ,i_l. 3
._---_, ,_..._
2._
_:._r!_. __
_; o.4o
'_t-
4O
6O
70
_
80
15
90
100
Figure
4.2.014
4.2.4.
Design
of
Isogrid
Cylinders
F
N a = _-_
M
=
Nb
300
-
-
0. 993
k/in.
=
1, 106
k/in.
27r(48)
8000
-
_
rr(48)2
N
= 2. 099
k/in.
cr
Since
, the assmnption
N b > Ncr(1)/4
1
Na
+Nb=N
for validity
of
E-':_
t*2
-
cr
_3
(1-v
10"6
=
2)
R
is fuJ filled.
cr
ER
From
the
2099
(48)
- 11
39.7(10
-7 )
graph,
ooo ,
Po
The
design
=
1.03
=
69.0
psi
is compression-crltical
weight
is obtained
is valid and,
tmin
(67.0)
b y using
= 0.00149(48)
:
=
10.3
> Pburst
and
the minimum
graph,
based
(F_u)104
> 55 psi.
since
Po"
Also,
0. 0715
in.
the
Po
(t--rnin/R)
N
-
ER\
= 3.97(10
Po]
"6)
_6_7.0 (10 -'3)
69.0
= 3.8_
4.2.01fl
(I0 "3 )
upon
Pc)
(103i
2. 505
X
y
0. 397
=
From
=
t
-
_
=
=
I
0.397
P
1.535
L
J
69.0
6 graph,
0. 273,
Po
Ftu(l
d
3.85
2.50_
9.42
thea,
a
-
6t
-- 13. 5
R
+_
=
13.5(0.
_d
b
q388)
=
h:
2
=
d._'5
=
10"3
0. 524
38.5 (I0 -4)
0. 616
t =
a.
69. 0 (48. 0)
67.0(1.273)(
"--
in.
(0. 524)
(1°2) 10.0388)
h = 2(2.00)
=
2.31
in.
0.0388
)
=
0.0791
(0. 524)
= 51.5(0.0388)
m.
a check,
bd
th
t
Worked
0.0415
(0.
O. 0_88
(2.00)
= t (1
¢ 30)
5,.24)
= 0.0388
Example
2
R
= 150
in.
F.
:
5 (10 6)
M
= 59.
10.
=
5 (i06
0.280
[
1
+
3
(0.280)
]
=
0.0714
psi.
) lb
in.
4.2.016
J_
0.0415
= z. OOtn.
I. 732
As
=
in.
in._"
Pburst
= 60 psi.
Ftu
= 71.0
Nb
M
59. 5 (106)
=_- _R z -
_(1502)
842
5 (150)
(10_6)
N cr
E-'-'R = I0.
From
the
ksi.
= 5. 35
= 842
lb/in.
(10_7)
In this case
=0.310
(71.0)
the cylinder
-cr
N - ER
Ft u ]
0. 000444,
Fo
--22.0psi
< 60psi
is pressure-critical
= 5.35
(I0"
=Pburst
and Pburst
= 6.33
\Pburst/
(103 )
""
59.5
(106)
70700
graph,
=
X
=
2. 505
0. 633
- ------- = 0.Z6Z
2. 505
iF,u0.633171o 1.89
_
Y - 0.397
From
graph,
_P--I=
0. 397
a = 0.0118,
= Ftu (I ÷ _)
_60.0 I-
5= 6.8
71 (I.011d)
d = 6t = 6.8 (0. 1252)
(I0"3) = 0. 1252 in.
: C. 852 in.
4,2.017
must
be used.
(10 -4 )
_ _
b=
h = _f10.2
_
d=
N
,[ 10.2
t =w6.3---'_
z
a -
As
_I6.33
_
(10 -4 )
0.616
h -
= 0.0321
(104 ) (0. 1252)=
z (15.9) = !8.38
_
(0 " 852)
127
(0.0852)
(0 • 1252)=
= 0.0273
15.9
in.
have
had,
in.
1. 732
a check,
bd
O. 0273 (0. 852) = O. 0117
t'-ff : O. lZ52 (15.9)
For
the
equivalent
T=t(l
weight
thickness,
+32)
= O. 1252
[I + 3 (0.0117)]
_" = O. IZ97 in.
If the
c,/linder
had
been
critical,
(15o)
t--rot n = 0.000444
: 0.0666
compressive
t,_.
4.2.018
one
would
in.
4.2.4
Summary
of Design
Equations
Co__m_ression
and Bending
co
c
1
cZ
for
Cylinder
Ref.
page
= 0. 397
4.2.
005
=
10. Z
4. Z. 006
=
0.616
4.2.008
Kef.
eq.
(4. 2.18)
N
N (103 )
(4. 2. Z3)
2. 505
Y
Under
=
N
0. 397
(4. 2. _4)
pR
(4. Z. Z8)
Ftu (I + a)
d
=
b
=
(4.Z. 30)
5t
(4. Z. Z0)
16
(4.Z. zg)
h
As
a check,
t"
=
t(l+3a)
4.2.019
Axial
4. 3
CYLINDERS
UNDER
TORSIONAL
SHEAR
1
The
cylinder
is
The
internal
shear
N
loaded
load/in.
T
-
by a resultant
=
cylinder
radius.
4. 3. 1
Typical
Design
Situation
occur
because
may
of spin
torques
be
to approximate
used
applied
to transverse
shear
erence
2-1
4. 3. Z
Method
The
which
and
all
optimum
loads
for
flight
the
required
stability
accompanying
that
the
opposite
ends.
equation,
control
purposes.
dimensions
bending,
shear
method
rib
is varied
optimum
of maneuver
two
buckles
The
around
References
are
located
fins
or
analysis
the
neutral
;)-13
and
in neutral
because
may
also
axis
due
_-14.
axis
rZef-
region.
of Optimization
optimization
buckling,
are
shows
by the
on the
cr
R is the
torque
given
, Nx_ , is
where
High
T,
V
Z _ R.2
x¢
torque,
pressure
designs
pressure.
assumes
crippling.
to obtain
which
whose
In the
general
instability,
skin
An
auxiliary
burst
pressure
is
introduced
minimum
weight
of the
design.
This
defines
into
t_vo classes.
divides
burst
simultaneous
thedesign
pressure
second
(which
class
are
4.3.001
may
be
alldes_gns
zero)
In the
is
whose
less
burst
an
firstclass
than
the
pressure
exceeds
the
optimum
be used
for
the
General
Stability
From
pressure.
In these
cases,
the
higher
pressure
must
Vcr,
is %iven
design.
Reference
2-6,
the
general
instability
torsional
shear,
by,
V
cr
(1}
(4. 3. 1)
----.
where
co
The
=
0.747_ 3/4
recommended
value
for
3/4
is
0.67.
Thus
co
Note
= O. 50
that E,:-"
and t:::
may
resultant,
V
be used
since eq. (4. 3. 1) is in terms
cr
Sub stituting
into
E:'.:=
E
t:::
=
t
eq.
(4. 3. 1),
l+a
4.3.002
of a stress
p5/4
c 0 Et
v
(1)
cr
(4. 3. Z)
-
Skin Bucklin_
The
skin buckling
k
equation
_ZEt
appears
in the form,
Z
S
cr
An
12(1
unpublished
k
%,
- v Z) h 2
investigation by B.R.
Lyons
=
35.0
for
clamp-=d
edges
=
Z0. 1
for
simply
supported
indicates that
S
k
edges
S
It will be assumed,
for the purpose
of the section, that the edge fixity is such
that
k
=
25.0
S
eq. (Z.4. 11)
then from
(Z) = t(l +@)
=
Vcr
_cr
c
1
E(I + @) t3
h2
where
C
l
"
25.0
IZ
(_ Z)
(8/9)"
=
23.1
Rib Crippling
From
eq. (2.4. 12)
2N
" _3 =4"_I
_Z
=
Nxy
- --- t(1
2
+o)
+a)
_Z
4.3.003
(4.3. 3)
x
From
eq.
v
(4.2.
1 3) one
then
obtains:
(3) = c 2 Et (I +_) b2
(4.
cr
3.4)
where
c
-
2
(0.616)
2
= 0.533
A
summary
of
the
critical
load
c o Et
v
(1)
,. IB514
-
cr
L
is,
)1/2
(1
4.3.5)
+ a) 1/4
t 2
v cr (2) = c 1
E
t (1
4.3.6)
+ a)-_
b 2
V er
(3)
= c2
To
these
are
par
amete
r.
=
Ftu
Define
Et
(I +a)
nowadded
the
(4 " 3 " 7)
V
burst
pressure,
pR
non-dime_,
sional
eq.
(4.3.6),
is
regarded
as
a free
parameter,
(4.3.9)
ER
From
which
(4.3.8)
t (I + a)
the
p,
(4.3.8),and
(4.3.9),
(4.
4,3,004
3.
10)
From
eq.
(4.3.7),
(4.
3.8),and
(4.3.9),
2
(4.3.
Multiplying
(4.
3.
10)
and
(4.3.
11)
t 2 b 2
1 c2
4
't
h2 d 2
(4.
From
=
ic2
eq.
(4.3.5)
and
(4.3.9)
Et
=_
V
_5/4
ER
cr
R
1/z
5/4
(1
• T)
+ _)1/_
or
_5/4
9/4(L)1/2 K
c o Ftu
Using
eq.
=
I0/4
(I +_)
(4.3.8),
/4
(1
c 0 Ftu
P
+ _)9/4
/
'_
p
(1 + a,) 10/4
4.3.11_
i
_
{35/4
10/4
(I 4_)
or
"_0
111
3.
1 2)
4/5
%f-_-
Now
P
Ftu
]
-
(4. 3. 13)
(1 + _)2
set
x
(4. 3. 14)
4/5
The
Y
=
x,
y; a,
equivalent
6
dependence
weight
is
thickness,
P
(I
+ a) 2
seen
to be
the
t-,
same
as
previous
graphs.
The
is
m
t
Using
= t (1 + 3a)
eq.
(4. 3. 8) this
becomes,
m
(4. 3. 16)
R
As
Ftu
1
in previous
cases,
P/Ftu
for
a given
value
the
of
quantity,
t'/R,
may
Vcr/ER
and
L/R.
be
minimized
as
a function
R
I
I
I
I
MINIMUM
iv)
4.3.006
WEIGHT
CUFIV§
'.
of
This
will
again
I.
For
divide
all
Pburst
These
_,.
In
the
compression-critical
Pburst
Po
classes:
Ftu
Ftu
cases.
For
this
case
be
the
design
is
may
nov¢
different
pressures-criticai
and
the
boost
pressure
and
associated
i_
shown
used.
curve
_,,IPo/Ftu_
values
for
Figures
4.3-1
and
complete
1.
two
Po_
Ftu
master
The
into
Ftu
are
must
A
<
designs
4.
and
x and
6,
Figure
t may
now
minimum
constructed
of
(t--rain/R)
Vcr/ER
and
L/R.
This
is
determined
by
the
following
y,
and
graph
read
in
3-Z.
geometry
Compute
be
from
the
procedure:
off
the
corresponding
o
4.3-3.
be
computed
weight
pressure,
from
the
Po'
burst
whichever
condition
or
is
larger.
from
the
pR
t
=
Ftu
(1
+a)
(4.3.
4.$.00"/
17)
01
30
0.15
: .-:
:_: ::_:
02
04
::::':i._=:
25
2(2
15
I
o 1
04
8 9 10
::i
'o.o
#
CR169
40
50
70
80
90
100
3O
SUMMARY
OF DESIGN
EQUATIONS
FOR
CYLINDER
UNDER
TORSIONAL
SHEAR
25
REF
PAGE
2O
CO
=
0.50
4,3,002
C1
23.1
4.3.003
C2
0.533
vc,
4.3.004
_ Er
(4.3.9)
13
=
_ _-/_
(103)
14.3,14)
3.51
10
9
F _ _141s
L_-_
Y
8
_
F,u
(4.3.15)
_7
pR
i4.3.17)
Ftu (1 + ¢l )
6
d
(4.3.18)
tit
5
b
= _
h
=F
_
d
(4.3.19)
t
(4.3.20)
AS A CHECK.
a
=___
"_"
=
th
t (1+3el)
I
4O
70
80
90
100
Figure
4.3-1.
4.2.001
m]tDou_vJ_,,_
X,Y,
a,
d Curves
for
Cylinders
Under
Torsional
Sheer
0
0
!
•
e_
::
::
:
i[] ::]!
;:: :iX
_D
0
i ....
v
T]I:
4'i,
:i ;h
iii ,i_i
!!il:ii
C
0
.ii
_"
o
o
H
4.3,009
8
i :
f ,
o
r,8.j
°d
4.3.010
.,.t
3.
Knowing
4.
t,
d
= 6t
The
rib
the
rib
depth,
d,
is
obtained
from
5.
(4. 3. 18)
width,
b,
is computed
from
eq.
(4. 3. 11).
(4. 3. 19)
5.
The
triangle
height,
h,
is
computed
from
eq.
(4. 3. 10).
(4. 3. 201
As
a check
value
of
Qf
on the
computation
reading,
m
with
the
value
4.3.
3
Worked
Example
R
=
48.0
in.
E
=
11.0
T
=
30
(106)lb
L
=
198
in.
read
off
from
graph
in.
= 4. o
Pburst
= 40.0
psi
Ftu
=
ksi
-
the
1106) psi
L/R
cr
graph
bd
th
=
V
and
a from
67.0
T
2_R 2
-
30
=
2075
lb/in.
2_
4.3.011
now
compare
the
computed
I
q
(
V
/
Z_cr
From
II.0 2075
{48.0)
=
(10-6) =
3.93
(i0=6)
graph,
/
P
o
_'_tu
(10-)
-
1.67
w
t
.Q
nlin.
The
R
=
0.00247
Po
=
1.67
design
tmi n
(67. 0) =
112 psi
>
40
is compression-critical,
=
0.00247
(48)
= O. 1186
in.
Vcr
( u)393
,06
u
V
= _
_
1.67
(lO -3)
2.35
(10 -3)
I
415
415
2.35
Y
0.50
=
14. 38
3.51
From
_,
6 graph,
a
=
0.285,
6
=
20.7
=
3.5---_
=
0.668
4.3.012
(2)
(I0"3)]
_
pR
(1 +or)
Ftu
d
= 6t
=
1.67
Z0.7
-
{10-3!
1.285
(0.0525)
=
(48.0)
=
1.290
O. 0625
in.
in.
b
Vb
=
O. 0858
v
h
As
O. 533
(1. 290)
in.
V
(0. 0625)
2.35
= 6.!9in.
a check
a
on the dimension,
= b._dd
th
=
"
0.0858
06 _ _
II
0
.
_
(6.
= O. 287
19)
m
t
Aa
= t (1 + 3a)
an additional check
=
6
From
=
0.0625
(1.862)
on all strength
= O. 1163
in.
calculations, from
the _ curve,
0. 287
= 20.7
=
24.0
eq.
(4. 3.5),
V cr
(I)
(4.
3.6),
and
(4. 3.7),
c o Et
=
2
0.50
(11.0}
(lO 6)
0.0625
(24.0)
- - (1.287)
(o.
o6z5)
514
48.0)
__
4.3.013
5/4
=
2030
Ib/in,
Figure
2-I.
V cr
(Z)
=
c ! Et
(1 +a)
(t h') Z
= z3.1 (11.
o_(io
6)(o.
o625)(1.287)
(°'°6z5)
z
6.19
V
cr
(3)
:
2080
lb/in.
=
c 2 Et
(1 +a)(_-)2
: o._
4.3.4
Summary
1.29
Ill. O__1o61Io o_,_1 (°'°8_8)
_ = Z080 lb/in.
of Design
Equations
for
Cylinder
Under
Kef.
co
c 1
c2
0.50
4.3.002
=
23. 1
4.3.003
= 0.533
4.3.004
m
-
Vcr
ER
V
X
page
=
Ref.
V
Torsional
=
eq.
(4.3.9)
(lO 3)
3. 51
(4. 3. 14)
415
(4. 3. 15)
Y
pR
(4.3.
-
Ftu(
i +&')
4.3.014
17)
Shear
d
=
b
=
6t
(4.3.
. 33
d
(4. 3, 19)
h
As
(4. 3.20)
a check,
t
18)
=
t (1 + 3_)
4.3.016
4.4
GYLINDER
UNDER
UNIFORM
EXTERNAL
PRESSURE
P
Q
.|
4--
-qP
441--
p
4-4-
The
loading
sure
over
of the
cylinder
side
walls
the
If N x is the
cylinder,
internal
consists
and
axial
ends
of a condition
of the
load/in,
and
of uniform
external
pres-
load/in,
in the
cylinder.
N_
is the
external pressure
and
internal
hoop
then,
N@
where
= pR
p is the uniform
R
is the radius of the
cylinder.
4.4. I Typical
The
most
Design
common
submersibles
Situations
design
or vacuum
situations for this conditiGn of loading
tankm
In some
bending
loads are
or more
commonly
small
pressure
loading.
In these
a "first cut" at the design.
to accommodate
4. 4. Z
The
cases,
small
the subsequent
Usually
only small
for
additional axial loads
superimposed
cases,
occur
analysis
upon the external
may
modifications
be used
for
are necessary
the additional loading.
Methc, d of Optimization
method
of optimization
assumes
simultaneous
4.4.001
failure for rib-crippling,
W
skin
buckling,
and
duced
and
mines
an allowable
is
designs
into
burst
pressure
sure.
In the
pressure
must
varied
two
instability.
to obtain
burst
second
minimum
may
class
zero)
are
all
cylinder.
minimum
tank
weight
class
are
all
designs
is less
than
the
minimum
whose
burst
weight.
designs
In these
cases
the
.
parameter
of the
first
be
pressure
weight
for
In the
minimum
A burst
pressure
classes.
(which
for
actual
,
is intro-
This
deter-
and
divides
whose
actual
weight
all
pres-
pressure
exceeds
burst
pressure
the
be used.
General
Two
general
.
Instability
cases
are
considered.
1.
The
"long"
2.
The
"intermediate
According
These
are,
length"
cylinder.
cylinder
to Keference
2-6,
the
intermediate
length
cylinder
lies
in the
range,
_0_0_0z
_. (_( :_
_/_
and
:, (10_/
sin ce
1-v
Z
=
_f_-
1+_
'
0.943
t
-
(4.4. l)
For
the
Ion
Reference
s
cylinder,
the
general
instability
Z-6.
3
Pcr
=
Co EI_-)
4.4.002
pressure,
Pcr'
is given
by
.
.
I
}
where
O. 90
cO =
=
=
O. 253
Thu s,
3
Ncr
(la)
=
Pcr R
=
Ncr
(la)
= c o ER(-R-
the
intermediate
c o E*R(-_")
t
For
3
)
length,
(4.4. Za)
--_1+_
cylinder,
Reference
2-6
co E
(÷)
L
where
0. 855
_
0. 855
.--
co
(0.75)
= O. 70Z
_--
T hu s,
c o E*R
N
cr
(Ib)
N
(lb)
=
=
Pcr R
=
(__)R5/2 (__)L
c o ER
(4.4.
cr
4.4.003
Zb!
Skin
Buckling
Tests
by
Jenkins,
curve
for
skin
Reference
buckling
is
2-13
linear
have
shown
that
the
in the
range
of interest.
biaxial
interaction
N x
URE
2
LOADING
!
N¢_
SKIN
From
the
N_
Thus
interaction
Z/3
INTERACTION
CURVE
diagram,
(External
taking
SUCKLING
Pressure)
of the
Z
= _- N_
(Uniaxial)
from
Reference
allowable
2-1Z.
t3
Ncr
(Z)
=
cl
= T
(4.4.
3)
c 1 E (1 + a)'_-
where
Z
Rib
Crippling
Eq.
(Z. 4. IZ)
1 ribs
has
the
(1°"z)
-
6.80
shows
that
one
in either
same
circumferential
rib
crippling
may
conservatively
or
allowable
longitudinal
as
for
4.4.004
set
Ncr
(Z)
direction.
unlaxial
loading,
=
Nx
In this
for
case,
the
one
!
z
Ncr
(3)
-
c Z E (1 ÷ _)(b.)
t
(4.4.4)
where
c2
-
0.616
B_.___t
Bur
The
burst
pressure
Collecting
is
given
Pcr R
=
by the
equation,
Ftu
=
pR/t
(]
+ a)
formula,
Nor
(lal
Ncr
(lb)
=
coER/_)
t
3
--_
1 + or
c o ER
=
Pcr R
(4.4.
[33/2
R .s/2
(1 +o)
(T)
1/2
(4.4.5)
t3
Ncr
(2)
N
(3)
cr
N
c I
(4.4.6)
E (1 t o)
bEt
Ftu
Define
=
=
c__ E (I +_)
-
pR
t (I +a)
the
auxiliary
=--
E --
variable,
(4.4.7)
7
(4.4.8_
_.
(4. 4.9)
#.4.005
5a)
From
(4. 4.6),
(4.4.8),and
(4.4.
9),
t2
N
=
From
c 1
(4.4.
(4.4.
10)
(4.4.
11)
(4.4.
1Z)
h2
7),
(4.4.
8),and
(4. 4. 9),
.m
N
Multiplying
(4.4.
10) and
= c 1
hZd 2
(4. 4. 1 1)
= c I
4
Z
(_)
clc2 _54
{y
The
_
c_¢rVS-V
positive
root
Equation
From
(4.4.
eq.
(4.4.
is taken
since
satisfies
skin
buckling,
(4. 4. 8),and
(4.4.9),
12)
5a),
_,
a,
and
5 are
rib
all
positive.
crippling
and
burst
conditions.
$
E
-
=
Co Ftu
(1 +a)4
.%_)3 2
•
(1 + a) 4
4.#.0_
k._
.__3
co
I_l
2
=
(l
I _o "iI z F t-u
P
Eq.
(4. 4. 1 3a)
From
eq.
satisfies
(4. 4. 5b),
_Z
+ _ )4
p
(4.4.
(1 + _)2
burst
and
(4. 4. 8), and
general
instability
,,u
(._..R)
5/2
a lon_
cylinder.
(4. 4.9),
CoER
Co
for
1 3a)
3/z ]
(÷
(I +a) I/2
But
R
T
F____.__
=
P
(1 + a),
Thus
or
z/3
(4.4.
41,4.007
1 3b)
I
Eq.
(4.4.
13b)
intermediate
satisfies
length
Equations
(4.4.
burst
(4.4.13a),
conditions,
moreover,
they
(4. Z. Z6) for
buckling
of the
may
be
general
instability
for
buckling
of an
13b)
all
instability
and
cylinder
12),
5 curves
an_
used.
and
are
(4.4.
seen
to be
cylinder.
Thus
As
define
the
satisfy
similar
to eq.
a consequence,
burst
(4. Z. 25)
and
the
x,
y; _,
(4.4.
14)
same
quantities,
and
1/2
1
=
(4.4. lSa)
Ya
Co
J
P
(1 + a) 2
or
2/3
(4.4.
Yb
=
to be used
[ cl-q
0 RL]
with the x,
F_____=
P
y; a,
5
(1 +a_)2
curves.
m
The
equivalent
weight
thickness,
t, is
m
t =
t (1 + 3a)
4.4._
15b)
Using
eq. (4.4.9) this becomes,
t
i
=
Ftu
_
I+ a
or
t
(] )
= t_ u
(4.4.
l+3a
As
in the previous
of
P/Ftu
cases,
the quantity t/R
for a given value of Pcr/E
and
may
be minimized
as a function
L/K.
T
R
_-
AND
L
I
I
I
°o
PlFtu
MINIMUM
WEIGHT
CURVE
If a family of {t--rain/R)
and associated
(Po/Ftu) valued
are computed
different Pcr/E
nondir,,ensional
curve
structed.
The
This
complete
1.
L/R,
is shown
geometry
Compute
and
and
x
6 , Figure
a master
aud
may
for
be con-
on pp. 4.4. 01g and 4. 4.0Z0.
is determined
y
and
by the
from
the
4. 4-1.
4.4.005
following
graph
read
procedure:
off the
16)
corresponding
015
01
02
30
25
20
0152
15
10
9
8
7
6
25
15
I
01
015
92
025
o 6
/
3
b
....
t;
• .....
o,+_
........
:L'-.i::F.!:!:i_)
2-L_
_:
'
iL;ii :
..... -.- ......
4,.
-L"
-- LJ-.;
......_,.,
-_.};
......
_,J......
: :;:|,,
_:
LL
"]!2
?
, ;T;;I ;
.++
-_."
:4
....
T'T
..... :-: 7: LT
.[!.iZ:..: :
=!i i-Li:i':l !_ _
.+':.+:'Z_
ZZ', ,_,f:
-_..,,.._,_I:_; 1 :;_-
': .......
t ,..._,Z:-r_.r_r
_ -_i....
£_? tt_
_i
;++LG
L+Z-.
+2
(J (32
[
Lml_:,,+,i
[.L: .IV
2
25
h
/
7
10
2O
25
30
7O
CR169
,
40
50
60
70
80
90
SUMMARY
OF DESIGN
UNDER
UNIFORM
100
EQUATIONS
EXTERNAL
FOR CYLINDER
PRES_>URE
REFPAGE
25
_L_
20
INTERMEDIATE
102_A
CYL
LONG
CYL
<_ 4 (103 )
A>4
X 103
Iit
15
Co
=
0.702
0.253
4.4.003
i_i
CI
=
6.80
6.80
4.4.004
;ii
C2
=
0.616
0.616
4.4.005
I;:
REF
10
-+-e-
EQ
9
l
Pc.
8
(4.4.9)
7
''_
I I
(4.4.14)
"_
II
Ill
III
X
=
(103)
2.04
6
( LONG
(4,4.15a)
CYL)
.+-4-4
(4.4.17)
PR
t
=
Ftu
(1
+ =
)
;.4-I-
(4.4.18)
m
2.6
d
=
_t
2
b
"
_
d
h
=
_
t
(4.4.20)
0.40
(4.4.19)
1.5
::1
AS A CHECK,
I
30
40
60
60
70
80
90
0'
=
bd
t--h'-
T
=
t(1+3¢1)
100
Figure
4.4,1.
X,
Y, a,lJ
_,=m=
4.4.010
lP0U)017
Curves
for Cylinders
Under
Uniform
External
Pressure
m
i
t
Z.
may
now
be
or the
burst
pressure,
Knowing
4.
the
minimum
whichever
is
weight
pressure,
=
The
larger.
(4.4. 17)
t,
and
6,
the
rib
depth,
d,
is computed
from,
(4.4. 18)
bt
triangle
height
is computed
from
eq.
(4. 4. 11)
(4.4. 19)
t
5.
The
Po,
1 +_)
Ftu(
d
from
pR
t =
3.
computed
rib
width
is
computed
from
eq.
(4. 4. lZ).
(4.4. Z0)
As
a check
on the
reading
of the
off from
the
4. 4. 3
Worked
Case
of Long
computed
graph,
x,
y;
the
a,
values
quantity
5 graph,
of
b,
d,
(bd/th)
Figure
t,
should
h
and the
equal
4. 4-1.
Examples
Cylinder
Pcr
=
600
pel
Pbur st
=
0
IA
=
10.5 in.
Long
E = 18.0 (106 ) psi
4.4.011
cylinde
r.
the
accuracy
value
of the
of
_ :ead
Per
T
From
(106)
long
600
18.0
-
cylinder
33.3
-
graph,
Figures
4. 4-2
and
4.4-3
D
t
R
=
PO
o. 016
(lO3) :
lO.7
=
0.016
(10.
Dcr
Ftu_
p I
F_
m
t
w
N
E
5)
=
0. 168 in.
33.3
=
(10 -3 )
10.7
=
3.11(10
_:_
]
-3)
llZ
Ftu
Ya
=
=
0. _53
10. 37
-
X
From
1. 523
graph,
=
6 =
0. 320
14.5
pR
C
10. 7 (10-3)(10.
"
"
Ftu (1 +_)
d
= 5t
=
14. 5 (0.0852)
1.320
=
1.235in.
4.4.012
¥
.'_
lmlmEpmm,,,,-w_-"
,0
,_o.-_-)
5)
=
O. 0852
in.
E
0
_1,,,
E
L.LI
E
4
e"
D
6
3
_or,
I.L
4,4.013
\
0
v
n
fl0
c-
X
UJ
E
r-
_t
_J
0
U.
4.4.014
r
i
d
b
h
As
=
_]3.11(103
0.616
=
6_ _
t
)
=
(1. Z35)
= _
3.11
6"80(103)
(1. 235)
(3.99)
=
O. 0877
in.
:
3.99in.
6250
lb/in.
(0.0852)
a check,
bd
th
_
=
t (1
+ 3a)
=
600
(10.
5)
_,
5,
_
graph,
w
t
Pcr
From
g
the
a
Using
eq.
0.0877
0. 0852
=
=
0.318,
6
(4.4.
5a),
(4.4.6)
0.0852
=
6250
(1.953)
6300
2=I,
14.5,
_
=
and
:
0.
1665
in.
lb/in.
Figure
(4.
=
18
4.7)
3__L_
l+o
cr'l) : 0ZS3
=
0.318
Ib/in.
t 3
N
cr
(2)
=
6.80
E
(1
+a)-_
(0
6.
80(18.0)(106)(1.
0852)
3
318)
(3.99)
0,0.0_
2
=
-I
I) 2
N
(3)
cr
0.61b
E
0.616
(18.0)(it)
6260
Worked
Example
0
=
R
:-
48.0
E
=
10.7
L/R
= _
1_2
Fre.,,
0877
0_.235
(1.318)(0.0852)
psi
Pburst
06
6)
lb/in.
100
(I
* o_) t d2
2
Pcr
Pc r
"-if-
(I
Ftu
::
67.
0 ksi
I_
:-
192
in.
(10 6 ) psi
:_
4 0
_.
)
-
I O0
10.7
3)
=
4.55
=
O. 0065
burst
pressure,
-
9.35
graph)
PO
--(lO
Ftu
m
t
...m
i__9.n
R
The
optirpurn
Po
The
=
4.55
minimunl
t-
m in
(67.
0)
equivalent
=
0.0065
(48)
=
Po
305
is,
psi.
weight
thickness,
=
in.
0.312
4.4.010
train,
is,
) 2
Pcr
m
N
m
4. 55
= _
X
I10-6) =
_ 9. 35
E
2.05
(10 `3 )
(10 -3 )
(,1o3) = 2.05
2.0"_'4" =
2.04
1.01
2/3
Yb
O. 702
2/3
[.O.o7.1°"4,]
From
x,
t
As
y;
o,
a=
0.27
6=
16.6
6 graph,
pR
"
= 4.55
Ftu
(1 + a)
=
d
=
6t
b
=
J-_OO_
h
=
._.80
_/
d
t
4.4-
1
(lO-3). 48. o
(0.
172)=
=
J2.05
=_6.8(;
2.85
= 0.1645
0. 172
in.
in.
_.616(I0 "3)
(103 )
2. o5
a check,
ae = _bd
O. 172
=
1.27
16.6
_
Figure
(2.985 _ = 0.275
(9.
4.4.017
(2.85)
(0. 172)
=
=
O. 1645
9.91
in.
m
t
=
t (I + 3a)
N cr
:
Pcr R
From
the
a =
Using
eq.
N
cr
=
13 graph,
Figure
0.275,
6 =
(4. 4.5b),
(l)
-
=
0. 172
100
(48)
(I.81)
=
--
4800
0.311
in.
lb/in.
2-1
1.6.6,
(4.4.6)
13 --
and
18.8
(4.4.7),
0.7025/Z ERL_ [ (1 +
133/2
c_
R
0.702
(!0.7)
(106 ) (48.
O)
(18.8)
5/2
(48.0)
O. 172
3/2
:
4980
]/z
19._.z
48
(1. 275)
(106 ) 1.275
(0. 172) 3
(9.91) 2
t3
N
(2)
:
c r
6 80E
"
6.80
(1 + a)-h2
(10,7)
:
4800
lb/in.
b2
Ncr
(3)
::
0.616E
(!
+ a) t d2
0 616
(I0.7)
(I06)
•
:
4820
(I
275)
*
Ib/in.
4.4.01M
o
&
(0
172) (0. 1645, 2
•
(?..85)
2
Ib/in.
Summary
_F_ternal
4.4.4
of Design
Free sure
Equations
for
Cylinder
Under
Uniform
Ref.
page
4.4.002
A
=
0. 530
1 +
Intermediate
102
_. A
co
=
<_ 4
1
c2
A
( 103 )
Long
CyI
>
(10
6.80
=
0. 616
3)
R.ef.
page
6.80
4.4.
004
0.616
4.4.
005
Ref.
q.
Long
Cyl
--
4
O. 253
0. 702
Intermediate
c
Cyl
Cyl
m
N
X
(4.4.
9)
(4.4.
14)
(4.4.
15a)
(4.4.
15b)
(4.4.
17)
(4.4.
18)
(4.4.
ZO)
Z. 04
1/Z
[ ]
_N__
0.253
(Long
Cyl)
(Inter
reed.
z/3
P
d
= bt
b
4.4.010
Cyl)
(4.4.
h
As
a check,
0t
_"
"
bd
th
= t(1
+ 3a)
4.4._
19)
4. 5
As
IN-PLANE
noted
in Subsection
2.4,
stresses
For the
in isogrid
case
of the
may
be
in-plane
plane
T
CONCENTRATED
stress,
the
Reference
LOAD
if changes
solution,
IN INFINITE
SHEET
of curvature
are
negligible,
easily
determined
concentrated
load
if N x, Nxy, and
for an isotropic
due
immediately
to Mitchell,
is
the
N Y are
sheet
known.
in
obtainable
from
2- 2.
¥
Q
0
X
Let
the
from
P,
point
the
is
The
O,
at which
the
stresses
and
make
an
angle
0 with
the
origin,
0,
is
resultants
are
given
(3 +v)
origin
applied
stress
at
= 4_P
r
cos8
[.
N
=
P
4_
cosO
r
F1 _ v.
2(1
=
"
4Psine
v
r
[1.v
+ z(1
Nxy
Although
N x,
Ny,
stresses
results
sufficiently
positive
+v)
sin
sin
(4.5.11
(4.5.
j
Ze]
closer
than
the
reinforcing
At this
point,
stresses
are
and
around
were
sheet
sy:nmetric
and
using
thickness;
not
for
the
too
however,
majority
no problem
around
The
solution
Strictly
speaking,
stress
function
an Airy
drastic,
is
finite.
attachment_
developed
constant
accurate
the
21
(4.5.3)
computed
skin
x axis.
20 ]
+v)cos
all
load,
28]
be
not
r
expressions,
4_.001
t
the
this
(4. 5. 1) to (4.5.3)
rotationally
along
concentrated
= 0,
eq.
for
directed
The
at r
to
coordinates
x axis.
infinite
used
ribs
the
be a distance
become
in the
size
computed,
Nxy
will
node.
to be
+ 2 (1 .v)
L
and
hole
is
by the
Nx
y
since
and
are
if the
it may
of design
be
the
will
in polar
thickness
variation
expected
to give
problems.
be
4. 5. 1
Main
Typical
Design
structural
loads
airload,
inertia,
from
the
mass
and
wall
or
it is desirable
may
be done
the
isogrid
plate
Typical
design
tanks,
tank
internal
of the
through
The
inertia
and
component
bending
parts
are
may
floors
stresses
loads
will
pipe
or
arise
from
gravity
thrust,
loads
be directly
For
weight.
tangentially
resulting
transmitted
bulkheads.
to minimize
transmitted
arise
from
equipment
etc.
In one
case
tangentially
into
the
supports,
transmitted
A-shaped
vehicle
to
most
cases
In general,
into
the
wall
of
shell.
baffles,
was
of the
intermediate
if the
or
shell
effects.
situations
a tank
by the
gravity
to avoid
this
on
carried
properties
shell
load
Situations
frame
in a strucbxral
test,
in interstages
(Figure
tank
by
Reference
between
4. 5-1),
means
a side
of an
2-12.
CR169
.mAJ*
F
P/2
Figure
4.5-1
The
reinforced
loaded
node
point
"'A-
Concentrated
No extra
A'"
Load
region
was
of view
prevent
a flat
the
A more
exact
shell
changes
of curvature
region.
and
to the
indicated
beyond
"beef
up"
load
analysis
was
This
as
well
using
set
analysis
for
bending
feathered
out
Fourier
of the
shell
wall.
Strain
gage
readings
This
was
analysis
showed
in this
tht,
section.
the
from
sheet
to
a flat
another
5 pockets
a double
to
purpose
reinforcement.
able
is
that
frona
from
served
local
series
away
tlowever,
for
material
to provide
of ribs
developed
point,
extra
as
first
this
in pz.rticular
in forming.
local
pocket
according
was
spot
the
loaded
first
of fabrication,
curvature,
to distribute
in the
designed
reinforcing
circular
of the
A_
P/2
SECTION
to account
sensitive
the
reinforcin_
for
to the
the
size
served
4.8.002
.........
"t
as
a "hard
pad"
total
pad
weight
The
cylindrical
loading
of
amounted
40,000
tc
involved
wall
with
loading,
i.e.,
20,000
only
100
lb/in,
less
did
not
Method
Since
v = 1/3
for
P
6_
..=
_
xy
Skin
Stresses
These
are
isogrid,
cose
r
y
entire
tank
was
reinforced
area.
The
any
side
a pad
loading.
diameter
an
axial
reinforcing
at
each
The
failure
load
was
an
axial
loading
of
for
no side
The
of
24 in.
lb at
each
pad
the
design
that
designed
penetrate
the
reinforced
eq.
1) to
(4.5.3)
and
load
for
pad
loading.
region.
of Analysis
P
6_
-
the
8-foot-diameter
5 lb
4.5.2
over
small.
to only
failure
S
of the
very
withouz
Moreover,
1_
was
load
lb in.
lb/in.,
Nx
the
2, 500
l o side
2,400
distribute
(-5
÷4
(4.5.
become,
sin E 0)
(4.5.4)
cos 0 (1 - 4 sin 3 e)
r
(4. 5.5)
P
6_
(4.5
sin
r
immediately
e
(1 + 4 cos 2 0)
obtainable
from
eq.
•
6)
(2.4.11)
?
(4. 5.7)
ay
= 6--_ rt
(1 + a)
1 - 4 sin 2 e
P
6_
sin 0
(1 + a)
(1 + 4 cos
E e)
rxy
along
load
Stresses
will
be
rt
a maximum
5P
ax,ymax
= ax
(_ = 0)
= " 6_
the
1
t (1 + a)
4k6._$
(4,
direction
w'._ere
0 = 0.
5.8)
1
Behind
the
a x (e
Rib
load,
where
= 180 ° )
--
Stresses
0 = 180
+
5P
6_
degrees,
1
(1 +a)
rt
(4.5.9)
(P in x direction)
2
V
Q
8
1,x
Rib
stresses
are
1
_r2, 3
9_
obtainable
from
eq.
8
9_
P cosec
rt
(I + a)
rt
(1 +a)
cos
+ 4 sine
cos Z{})}
(2.4.
12)
(4. 5. 10)
8 - 4 sin z 6 cos0
+_]_
(4. 5. I I)
(sin0
Rib Stress
(P in y direction)
V
2
p
RIB
Q
STRESS
(P IN
4.5,004
V DIRECTION)
In this
case
N
_
x
exchange
of x and
P cos e
6v
r
(1 - 4 sin 20)
(4.5.
(-5
(4. 5. 13)
P
6_r
cos0
r
Ny
-
Nxy
= -
P
61v
Note
that
8 is
,_ow measured
Eq.
(2.4.
12) now
gives
4
= "9
P cos 0
.'rrt (1 +_)
= 9_rt
P
(1 +a)
Grl
_Z, 3
sin
y is
eq.
(4.5.4)
to {4.5.6)
gives
+ 4 sin 2 01
0
12)
Z
r
(1 + 4 cos
from
0)
the
(4. 5. 14)
y axis
toward
the
x axis.
(1 - 2 sin20)
[-5
(sin 0 + 4 sin0
cos0
(4.5.14
+ 4 sin 2 O cos0
v %/_
cos z O)}
(4.5.15
Recommend
The
Design
maximum
Procedure
rib stresses
willocc_zr
for the l ribs for loads in the
x direction.
P
(1 +a)
:
eq.
(4. 5.9)
the
maximum
x,y
max
-
5
6n
rt
From
q
As
8
9-'-_" rt
_1 max
previously
manner
explained,
(4.
skin
stresses
are,
P
(1 +a)
(4. 5
the reinforcing
that the reinforcing
5. 16)
of ribs and skin will be in such a
is rotationally
4.S.006
symmetric
with respect
to the
17)
loaded
point.
saY.o,
It is
in the
further
recommended
unreinforced
region
be
of skin
and
that
the
maintained
nominal
value
of o,
in the
reinforced
region.
Since
8
9Tr
0.283
-
\
and
ah
o
where
= b._d A._6_
t
t
A
then
=
bd
=
a conservative
t
A
req
req
=
rib
area,
sizing
ribs
will
be given
by the
equations,
O. 283
Ftu r (I +So)
= a
o
(4. 5. 18)
ht
(4. 5. 19)
or
A
:
0.283a
req
Eq.
h
o
(4. 5. 18) and
in either
x or
to
Ftu
(4.5.20)
will
y directions.
r,
the
reinforced
loads
are
applied
regions
as
P
r (1 +_o)
in Figure
satisfy
Since
and
(4. 5. 20)
dimensional
tre q and
additional
4.5-1,
where
4.11.0_1
i
_dr
Are
requirements
q are
weights
some
inversely
are
yery
curvature
for
loads
proportional
small.
If the
changes
may
be
expected,
the
increase
in dimension
required
by
region
than
that
reduce
the
curvat_,re
changes.
It may
be
necessary
to check
using
the
equations:
be
(4.5.
may
18)
and
If this
be
"feathered
(4.5.20)
out"
to "spread"
is done
the
additional
skin
buckling
or
over
a larger
the
load
and
weight
will
still
small.
a (skin
allowable}
_. (rib
allowable)
4.5.3
Worked
for
local
=
10.2
(4. 5.21}
E h2
= 0.616
E
b2
d--_
i 4. 5
Example
2k/"
a
o
=
/
/
:
g
30k
= 0.31
h
= 4.1in.
Ftu
= 61.0
ksi
a h
= 0.31
(4,1)
0
0.283
P
Ftu(1
+ a o)
=
crippling
t2
m
I
P
rib
0.383
61.0
1.27
(30)
(1.31)
A
= N
t
:
0. 116
4.6.007
in.
2
2 _.}
Rib
b
4
On the
skin
= 1. 365
4.1
sh
actual
buckling
Skin
r
0.0988
a
._ = 1.365
0. 0329
c
4
-_h=
necessary
to check
the
according
to eq.
h
h
a
A
=
and
it will
rib
be
crippling
4_.m
4rr
0.0194
5.47
0. 0247
5.47
design
0.0778
final
{4. 5. 21)
dimensions
and
for
(4.5.22).
%
4.6
IN-PLANE
CONCENTRATED
4.6.
I
Design
Typical
Typical
design
and
are
l::edorninantly
4.6.2
Method
local
For
the
case
of
sheet
in
from
Reference
EDGE
OF
of
loading
occurs
where
this
type
load
due
to
edge
loadings
of
AT
SHEET
Situations
situations
attachments
LOAD
of
this
engine
thrust.
are
Many
interstage
connections
type.
Optimization
the
concentrated
generalized
plane
2-2.
It
in-plane
stress,
should
be
load
at
the
the
solution,
due
noted
that
eccentricity
edge
to
of
an
Flamant,
of
load
isogrid
is
obtainable
is
not
included.
Let
the
point
the
origin,
P,
is
which
the
stresses
and
make
an
angle
0,
applied
From
O at
at
the
Reference
N
=
2-Z,
2 P cos0
X
N
_
the
stress
cos
and
is
the
directed
- resultants
2
0
be
computed,
x axis.
along
be
a distance
The
concentrated
the
positive
r from
load,
x axis.
are,
(4.6.
!_
2 P
cos0
sin20
14.6.2_
nr
=
xy
0,
0 with
to
_r
y
N
origin,
are
_
sine
cos0
,4.6.
nr
4.0.001
'
Although
N x, Ny and Nxy
evaluated
closer than the reinforcing
point all stresses
skin around
ness,
infinite at r = 0, the stresses
are finite. The
the attachment.
it is probably
around
good
will not be
the hole at the node.
At th;s
solution will be used to size ribs and
Although
a very
vided that the reinforcing
half circle,
become
the solution assumed
approximation
does not depend
r = ri, the reinforcing
upon
is constant
constant thick-
for reinforced
sheet pro-
the angle 0, i.e., at each
around
the half-circle.
Skin Stress
The
skin stress is immediately
obtainable
cos
I
eq. (Z.4. II).
e
2/
sin 2 e
Z P cos{}
_r t (1 +a)
_y
from
(Tx
(4.6.4)
sin {}cos 8
_'xy
Rib
Stress
The
rib
or
in the
(1-Bars
in x direction)
stresses
will
depend
upon
whether
the
1 bars
are
in the
x direction
y direction.
P
\\\\\\\_
v
Y
O
BAR
LOADING
From
eq.
o"l
(2.4.
ORIENTATION
12),
2 P cos 8
3nrt (l +a)
(4.6.51
( 4 cos
4.8.002
_2,3
= 43_rt
P sin(iO+a)cosO
Rib Stress
(I = Bars
(sinO±
%/_-cos8)
(4.6.6)
in y direction)
x and y in eq. (Z. 4. I Z) must
In this case the subscripts
be interchanged.
P
,\\\\\\\\\\q
'\
Q
BAR
_r
ORIENTATION
ZPcos8
3,rrt
(1 +a)
--,
1
t OADING
(4 sin 2 8
4 P sin (} cos(}
3_rt
(1 +a)
#T
v2,3
(cos
{) ± _/3"-sin
Recommended
Reinforcement
The
value of the stresses
maximum
not exceed
cr
the value, _,
from
- I)
computed
(4.6.7)
(4.6.8)
e )
from
eq. (4.6.4)
to (4.6.8)
the eq.
ZP
=
_rt
(4.6.8)
(1 +a )
It is recommended
thata
will
be held constant at the nominal
value, a
O
Then,
P
tre q
=
0.637
_'tu
(4.6.
r (1 +a o)
4.1.003
q_
_-=_:
-_
and
A
th
=
u
o
where
=
A
bd
=
rib
area.
Thus
P
Are q
=
0.637
(4.6.
If P is a compression
load it will be necessary
rib crippling and skin buckling
using
to check
the ribs and skin for
the equations,
t7_x
(skin allowable)
:
(4.6.
_rib (allowable)
•
X
=
is computed
from
(4.6.4)
value
¢.
dependent
upon
the rib orie1_tation.
load direction,
x.
4.6.3
Worked
Example
=
67.0
ksi
=
3.5
=
0.30
Ftu
h
eq.
eq.
maximum
1
(4. 6. 12)
0. 616 E d-_-
from
(4.6.
and
5) and
I Bars
_rib
(4.6.6)
Maximum
is
computed
or
(4.6.7)
rib stresses
parallel to edge.
in.
0
4A.004
L
11)
I0.2 E-_
b _-
where
10)
_o h Ft u r (1. + Oo )
from
and
the
(4.6.8)
will lie in the
i
•oh
From
=-
eq.
A
t
=
(4.6.9)
0,637
P
0 " 30
(3
" 5)=
1 • 05
and
(4.6.
_
0.637
(40.0)
67.0
(1.3)
in.
10)
-
D
in.
2
+_o)
No.
Pockets
r
t
1
3.5
0. 0837
0. 0880
2
7.0
0. 0420
0. 0440
3
I0.5
0.02C0
0.0293
4.0._
I
O. 293
Ftu(l
A
=
1.05
t
>-
4.7
CUTOUT
REINFORCEMENT
4.7.
I
Design
Typical
Cutouts
are
provided
to match
the
grid.
Situations
for
access
and
It is assumed
are
that
assumed
to be
plate
is uniaxially
the
hexagonal
in shape
loaded.
2
\
\
/
T
/
3
CUTOUT
Although
the
concentrate
nodes
grid
bars
be
of the
order
of secondary
for
stress
Kirsch
Reference
be
noted
is inefficient.
these
provided
if circular
that
when
holes
the
use
doublers
it is more
the
pattern
around
the
stress
concentrations
stresses.
The
around
a circular
hole
of rectangular
or
square
will
at the
used
the
in generalized
are
to
redistribute
required
cutguts
to use
the
hexagonal
by isogrid,
stiffening
these
ribs
if required.
are
the
material
needed,
making
a hexagonal
sible
tearing
of the
4.7.
2
The
Kirsch
cutout,
unsupported
skin
should
to prevent
stress
skin.
of Optimization
solution,
hole
analysis
effective
again
Method
the
and
bar
2-2.
Large
cutouts
already
the
at the
stress,
It should
hexagonal,
forces
ofG.
plane
is
the
should
solution
hole
IN ISOGRID
in polar
coordinates,
4.7,001
is
in isogrid
the
load
rib
pattern
be removed
concentrations
around
In additions,
to the
and
rib,
pos-
COORDINATES
=
rr
--
2
1
2)
-
+ _
r
_
1-4
_
cos
1,3-
r
_q
=- 2
Cro
where
• is the
In order
to apply
cos
3
a,)
O"
1+
LOADING
2 0
(4.7. I)
r
ff
=OO--
AND
20
(4.7.
z)
r
1 + 2a__
2
r 2
-
4)
3 _-_
r
nominal
stress
the
solution
and
sin
20
a is
the
(4.7.3)
hole
radius.
to isogrid,
it will
be
coordinates
to obtain
(4.7.1)-{4.7.3)
in cartesian
Consider
the rotation of axes,
necessary
the
bar
to express
stresses.
Y
O
#
_x
COORDINATE
4.7.002
AXE8
Define
the
direction
i
a.,
J
cosines,
r
a
=
cos
0
=
sin
e
=
c
x
r
a
=
s
Y
(4.7.4)
¢h
a v
=
-sin
=
cos
6
=
-s
X
a
a v
8
=
c
Y
The
transformation
ff xx
equations
r
r
:
ff rr
a x
a x
=
ff
C
=
r
r
ff
rr a x a y
--
(=rr
-
=
(rrr
r
r
ay ay
:
_rr
s
+
are:
a
r a 0
x
x
SC
+
_r{}
+
aO r
Z
ffxy
_y
rr
a
The
define
the
eq.
2
¢rr
(4.7.
¢
Z
+
_eo)
+
8
e
a
x
x
_rO
_88
_r8
r a8
ax
y
+
sc
+
(c z
_rO
r a 8
ay
y
ere
+
s
(4.7. 5)
_Or
-
_Or
8
r
ax a y
+
_OO
8
O
ax ay
S Z)
O r
ay ay
(4.7.6)
+
_60
8
O
a y ay
2
+
2 _re
sc
+
(4.7.7}
_OO c
quantity,
(4.7.8)
1)-(4.7.3)
=
a
aOO
I
_r
--
+
Z
-
2
Now
aO ar
x
x
(1-
_Z)
become,
+
(I-
4_
Z + 3_4)
(c z . sZ)
4.7.003
.J
\
2
_OO = (1 + _2)
2 _ro
:
Multiplying
- 3 _ 4) (2 sc)
" (1 +2_2
by the
2 N
T
x
(1 + 3_ 4) (c 2 - s 2)
_
=
plate
2 + _2
thickness
and
using
(1 - 6 c 2 + 16 s 2 c 2)
16 c 2)
2 N xy
"T
=
_2 sc
-2' N
T
y
=
_2 (3 -
2c 2 -
where(2/v)
axx
=(2/T)Nx,
etc.
T is the
nominal
stress
resultant.
(6 -
-
(4.7.5)-(4.7.7)
+
12 _4 sc
16 s 2 c 2)
+
_4
N X is stress
these
become,
_4 (3 - 24 s 2 c 2)
(4.7.9)
(1 - 2 c 2)
(4. 7. 10)
(_3
+ 24
s2c
2)
(4.7
resultant around
11)
the hole while
Skin Stresses
The
skin stresses
from
eq. (4.7.1) to (4.7.3)
into eq. (2.4.11),
a
x
=
are immediately
obtainable
expressed
from
eq. (4.7. 9) to (4.7.11) or
in stress resultants
by substituting
e.g.,
N
N
x
t (1 +a)
y
t (I +a)
'
_ry
-
N
'
vxy
xy
= ' t (I +a)
(4.7
Rib Stresses
The
rib stresses
for the 1 bars in the x direction
eq. (2.4. 12).
_1
:
1
3 t (1 +a)
a2
=
3t
2
(1 +a)
(3 N x - Ny)
(Ny
,
I/3Nxy
)
4.7.004
are obtainable
from
12)
°3
If the
2
3 t (1 + a)
=
1 bars
are
(4.7.
(Ny
in the
-
y direction,
Recommended
Reinforcement
Solution
1) to (4.7.3)
(4.7.
thickness
using
polar
good
approximation
does
not depend
was
exchange
obtained
coordinates.
to the
upon
13)
_/_Nxy)
x and
under
One
reinforced
the
may
plate
y in eq.
(4.7.
as surz/ption
13).
of const_'nt
expect
this
solution
provided
that
the
plate
to give
a
reinforcement
0 and that the variation in the r direction is not too
rapid.
For
this
purpose,
stresses
are
reinforcement
a maximum.
This
Along
this
line,
s = 1 and
Along
the
y axis
(4.7.9)
=
2+_2+3_
N
=
0 (Principal
2
Skin
xy
= 392
Ny
Stress
along
-
will
be
computed
be
along
the
along
y axis,
the
Refer
line
where
ence
2-2.
c = 0.
become,
4
stress
requirement)
3g 4
(4. 7. 14)
y axis
°x
-
2t
T
(1 +a)
(2 +
ay
=
2t
(1 +a)
(
=
0
v
will
to (4.7.11)
2 N
T
x
2
.
•
the
f2
"
(4
+ 3g 4)
'
7
"
15)
)
xy
4.? .006
.....
!
At the
edge
(y
of the
t(!
+a)
v
= 0 (Boundary
=
y
1 Rib
Stress
From
eq.
xy
along
(4.7.
_
edge
ff
13) and
of the
It is recommended
required
skin
t
A
hole,
thickness
a be
held
constant
= rib
area
--bd
:
th
(2 • _2
+ 3_4)
A
th
m
a
17)
(4.7.
18)
at
nominal
value,
a
o
The
is thus,
T
(1 + a o)
=
(4.7.
14)
_ = 1 and
2 Ftu
_o
whenA
(4.7.
(1 + Z_ 4)
that
:
req
16)
condition)
3T
t (1 +a)
1
(4.7.
y axis
T
t (1 +_)
al
At the
_ = 1 and
3T
_-
x
hole,
O
4.7.00e
(4. 7. 19)
or
A
ao h T
2 Ftu (l + _o I
=
req
Equations
(4.7.
19)
arid
(2+_ 2 +3_ 4)
(4.7.20)
can
be
(4.7.20)
simplified
if one
notes
that:
T
o
Ftu
(1 +_ )
treq
=
to (1 + ._
1
then
and
(4.7. 21)
2 +_4)
3
since
A
A
=-- t
t
=c_oh
0
I
A
o
req
=
4.7.3
Worked
A
_
(1+
(4.7.22)
+ __
1
2
3
4)
k/
_
I
Examp!e
_t
v
%t
.
v/k/k_ ,/V_ ,X,,'v
V;,,A
'k/k/k, _.
V/X/%
,/k/hA
\/
.,_
/k/ X/v
vV',,/VVV
'_/VVVV\
,AA
/\
r%
4.7.007
•
_a
!
Let
the
The
c and
The
skin
and
the
b
bars
are
in the
first
The
skin
and
the
d bars
are
in
the
second
The
skin
and
the
f bars
are
in
the
third
set
For
the
skin
and
skin-related
bars,
b,
d,
will
be
calculated
at
the
radius
Assume
a bars
be
e bars
tion
of the
The
following
and
(4.7.22).
hole
the
bars
are
parallel
1/3
around
the
to
of
the
the
equal
edge
of
the
hole,
of
triangles
a bars.
set
triangle
to
the
and
t/t
set
of
around
triangles
around
of triangles
and
f,
the
the
around
distance
hole.
the
hole.
the
hole.
from
the
hole
in
the
direc-
(4.7.
21)
height.
triangle
height
and
the
ioad
1 bars.
ratio_
of
A/A
o
Bar
_
A/A
a
1
b
o
_ay
o
be
calculated
from
eq.
Skin
_
t/t
3.0
b
3/4
1. 745
3/4
1.745
d
3/7
1.142
c
i/2
1.219
f
3/10
1.046
d
3/7
1.142
e
I/3
1.075
f
3/10
1.046
4.7.@_
o
4.8
OPEN
The
open
may
be
ISOGRID
isogrid
from
routing
of control
lines,
etc.
90-degree
grid
pattern,
0-
stable
and
possesses
ing
it as
and
For
a remarkab.:e
strain
properties
a sheet
of solid
Poisson's
ratio,
example,
v =
1/3,
t
--* 0.
when
for
weight
Lira
t --0
Eq.
are
of fluid
or
air
is
structurally
isogridpattern
of torsional
immediately
with
flow
alone.
Young's
This
or
rigidity.
obtainable
either
modulus,
E,
by
given
considerby
stress
or
by
considering
the
limiting
cases
1)
in
calculations,
/
bd\
bd
t-_0
\
"/
h
+
3_):
(4.8.2)
calculation,
t-:
(4.8.2)
for
(4.8.
formula
for
the
degree
material
t _0
and
of free
of ribs
b
_
= E
previous
the
of a gridwork
of view
the
stress
to
consists
point
Unlike
The
WEBS
construction
desirable
access,
E
SHEAR
Lim
t -0
may
be
(t
used
in eq.
3 b_d
h
(2.4.
12)
(4.8.3)
to obtain
h
_1
= _
¢r2,3
=
the
stresses
in the
ribs.
(4. 8 4_
(3N x -N y )
2h
3 b"-_ (Ny
•
4,8.51
'_'Nxy)
4.8.001
The
twisting
strength
(2. 2. 5) and
of isogrid
3
=
be
calculated
from
eq.
(4.8.
Ebd 3
h
=
12 (1
The
may
1),
(2. 2. 10).
E'-d3
D
plate
-
32
v 2)
Mxy
I
= _ D
Mxy
1
32
Ebd 3
h
bending
and
extensional
(4.8.6)
(2Xxy)
(2×xy)
(4. 8.7)
stiffness
of open
isogrid
has
been
verified
such
by
test.
4.8.
1
Typical
This
sort
of design
wing
spars,
girders,
wall,
is
used,
it is
frequently
_lsually
have
to be
Isogrid
provides
required
redundancy
rigidity
Design
and
Situations
situation
usually
occurs
for
webs
of beams,
etc.
In some
cases,
the
entire
panel,
such
When
solid
members
shear
web
by
to carry
the
shearing
necessary
reinforced
for
forces.
to penetrate
around
great
the
flexibility
in case
of damaged
strength,
it can
for
the
edges
in hole
location
members.
easily
to prevent
as
Because
accommodate
edge
well
of its
as
as
a
are
holes
which
crippling.
as
high
unanticipated
considerable
twisting
wracking
loads.
Open
grid
construction
spar
webs
when
floor
beams
where
the
fu3elage
{Figure
multiple
Apart
situations
penalty
engine
from
the
fuel
as well
containment
cables,
general
one
and
in large
need
a structure
as
webs
would
is not
a concern,
for
transport
and
ventilation
ducts
are
wires,
4. 8-1),
clusters
require
shear
with
for
for
space
systems
for
wing
rib
aircraft
routed
distributing
and
down
thrust
boosters.
structural
which
frequent
beam
be useful
can
efficiency,
these
be penetrated
without
excessive
attachment
of
opportunities
4.8.00_
for
design
from
CR 169
NXy
IL
1
d
, vv,2I!
.
_J_
_
I 4
f
_
-T
v
X
b
= !
Nxy
i
Figure
4.8-I.
Shear
Panel
support
brackets
desirable
for
Support
beams
requiring
This
or
free
flow
several
4.8.2
stiff.
Methods
The
ventilation
supports
machined
method
of
long
along
minimized
Both
of
construction
purge
gases.
spans
also
tend
span
to
prevent
the
beam
if
these
isogrid
Open
or
the
be
open
of
components.
over
can
torsionally
integrally
of
extending
requirement
and
equipment
is
characteristics
also
laterally
rolling
unstable,
under
load.
a symmetrical
section
inherent
in
panels.
Optimization
optimization
assumes
that
are
equally
instability
buckling
of the
Under
these
ass,,mptions
the
weight,
plate
spacing,
see
Figure
4.8
b,
and
grid
spacing,
depth,
c,
and
integral
width/grid
ribs
general
column
= rib
be
are
Euler
_-b
to
is
likely,
thickness,
l,
and
in-plane
see
Yigure
4.8-1.
d,
and
ratio,
are
the
determined.
a
The
magnitude
satisfy
_
and
make
height,
h,
and
so
that
plane
of
the
plate.
of
the
of the
rib
the
width,
plate
d
> b to
prevent
Etller
4.8.003
column
a,
are
determined
to
multiple
of
the
triangle
buckling
of the
ribs
out
Consider
a
General
Instability
From
rectangular
reference
plate
2.9
for
of
gross
length
k
f,
buckling
n2
_
width
of
the
c and
thickness
plate,
Ncr(1),
d,
d3
14.8.
8
Nxy
where
k
:
Nc r (])
_s
the
:
8)
12( 1 - v z) c z
buckling
coefficient
of
the
plate
which
depends
upon
c/f
and
s
the
plate
Using
the
boundary
conditions.
effective
modulus
3_r 2
Nor
(1)
-
32
E
Ebd
ks
and
Poisson's
ratio
from
(4.8.
1),
3
c2h
(4.8.9)
Ebd
=
1.10k
3
s
ac
since
Now
define
b
{3 .-- --
(4.8.
I01
(4.8.
11}
a
N
(1)
cr
:
1. 10
ks
d___._
E 0 3
2
C
In-Plane
Euler
From
eq.
Column
Buckling
=
¢2,3
=
±
P2,3
=
±
The
P2,
3
0
±2h
will
Rib
(4. 8. 5)
¢1
P3
of
be
Euler
%r_ bd
a
Nxy
=
(4.8.
1 2)
b_
Nxy
critical
for
compression.
column
load
for
P3
is,
2
k
p
EI
E
-
3
_
2
a
where
k
is
the
column
fixity
for
in-plane
buckling.
C
Thus,
k
_2
Edb
3
k
C
=
Z
0. 822
Edb 3
C
Z
IZa
or
using
"' =aN
xy
a
(4.8.
I0),
13)
(4.8.
Nxy
= Ncr(Z)
=
0.8ZZ
kc
Edf_
I
/
Equating
(4.8.
I. I0 k
8
1 3)
_
d
--
and
(4.8.
-
0. SZZ
/
1 1),
3
Z
k
d_ 3
c
¢'
C
¢
/"
4.$ .006
]
!
1
"
10k
s
k
0. 822
=13
2
2 c
-= 1.34
dZ
k
s
-k
C
i.
c
ee
d
_=c
(4.8.
1.34
J
14)
kk
C
Substituting
:
Nxy
(4.8.
14) into
1. 10
ks
(4.8.1
E -$
1),
.3 ._s
c
k
C
N
k3/Z
xy=
Ec
1. 272 (d;
s
C
Solving
for
d,
d093
i :cl/211/4
3/z
(4.8.
c
15)
8
Since
c
is given
from
d/c
and
The
equivalent
bd
_ = 3-g-=
and
eq.
(4.8.
weight
zC'T
d is
determined
by
(4.8.
15);
_
may
be
computed
14).
thickness,
_,
is
determined
using
eq.
(4.8.
3) and
¢_ .
(4.8.
16)
b.._d
a
4_.00e
B
•
t
Since
both
4.8.3
_ and
Worked
N
d
have
been
determined,
_
has
E
= 10.5
been
determined.
Example
= 8000
lb/in
(106)psi
xy
c
= 20 in
d >>
c and
From
simple
Reference
k
support
boundary.
2-4.
= 5.35
S
Since
eq.
(4.8.4)
and
2 bars
have
tensile
assume
that
the
bar
=
12.40
=
1.414
(4.8.5)
loads
show
equal
fixity,
that
to the
k c is
the
compressive
2.0.
Thus
k 3/2
S
k
!,2L
C
k
c
I/2
1.414
3/2
k
:
_
1
- 8.77
S
From
eq.
(4.8.
1.72
15),
\
Ec
1 bars
c
1/4
c -- 0 548[8°°°1_°-6_]s
L-_... 12o_
o : 4.30I,o-Zl
4.8.007
are
loads
unloaded
in the
and
the
3 bars,
d
From
= 0.860
in.
eq.
14),
(4.8.
_: __o,_o_,/_ _ (_?o
_)
From
= 0.0814
=-
eq.
16)
(4.8.
= 3.47
As
b
a
(0.0814)
a check,
from
Ncr(l)
= 1.10
(5.35)(10.5)(106
= 8000
Ib/in
From
eq.
N
(4.8.
C r(Z)
eq.
(4.8.
dimensions,
4h
= c
= ZO in.,
h
= 5.00
2
h
) (0.0814)
(I06)
(0.860)
Ib/in.
b/a
----
in.
11),
= 0 8ZZ (Z) (10.5)
the
a
= 0.243
try
4 pockets.
in.
= 5.77
in.
VT
b
= Da
b
= 0.0814
:
( 0.202860)
13),
= 8000
For
(0.860)
(5. 77)
0.473in.
4.8.008
(0.0814)
3
i/
Since
b/d
= 0.473/0.860
buckling
6f the
bars
= 0.551,
out
of the
this
plane
should
of the
be
figure
sufficient
to prevent
between
the
nodes.
I.
_y
For
a weight
comparison
with monocoque
construction,
for a monocoque
d,
o
Ed
Ncr(1)
-- 0.924
3
o
2
kI
c
10.5
8000
= 0.924
(106) (do 3)
(5.35)
202
].2.98
d
=
3/
o
The
d
o
3
8 (103)
o. 129
relative weights
t
T
o
The
(I04)
: o. 395 in.
are,
O. 243
-- O. 39------5- 0.616
isogrid
shear
panel weighs
a solid sheet neglecting
nodal
approximately
In the case of a built-up tension field beam,
strength of the material
(about Z0 percent)
caused
For
where
F su
web
-': 46, 000
percent
of the weight
of
weight.
the ultirrlateshear
a 7075-T6
61.6
by penetrations
pei
4.11.0_1
the web
corrected
gage
is determined
for loss of material
for attachment.
by
N
t
xy
= Fsu
w
If the beam
=
= 0.217
the
is efficiently designed,
is approximately
Applying
8oo0
(46000)(0.8)
ratio
of stiffener
weight
to web
weight
of r_:aterial
for
0.35.
this correction,
= 1.35
t
= 0.293
W
For
this case,
_'
t
then, the relative weights
O. 2.93
O. 395
= 0.7i2
weights
of the
are,
0
Relative
3 shear
web
Shear
Ttesistant
1. 000
Tension
Field
0.74Z
Open
If the
are:
O. 6 1_',
Isogrid
tension
constructions
field
design
were
";o be
an integrally
evaluated
without
stiffened
structure,
less
attachment,
that
is,
as
theoretically
be
only
0o 593
times
that
of a shear
resistant
web.
shear
resistant
because
the
integral
constructions
need
to be
tension
field
at the
-_tiffener
concentrations,
In any case.
suffer severe
wrinkles
particularly}
as mentioned
weight
risky
produces
when
u_desirably
life
previoL, sly, the continuous
penalties when,
inevitably be penetrated
fatigue
its
4.8.010
would
However,
interruption
high
locai
of
stress
is a consideration.
shear
as internal structure,
for useful service
weight
web
designs
they must
in a real-life vehicle.
4.9
OPEN
ISOGRID
CYLINDERS
LI
IN COMPRESSION,
BENDING
3
/
>
The
open
isogrid
cylinder,
loaded by a resultant force,
cylinder.
N
The
-
x
F
ZrlR
max._.mum
N
The
The
F,
of a gridwork
M
+
--
M,
skin, is
at the ends
of the
N x, is given by
cos
_-_
value
of ribs without
and resultant moment,
internal axial load/inch,
(Max.)
x
consisting
of N
F
2IIR
x
occurs
for
_ :
0 degree.
M
+ _u
HR 2
elastic properties
of the gridwork
sheet of solid material
with Young's
are obtained
modulus,
by considering
]_, and Poisson's
it as a
ratio, v,
given by:
v
For
:
1/3
(4. 9. 1)
flanged
individual
isogrid,
modulus
is
each
flange
given
by
and
(4.9.
The
rib stresses
For
the I ribs in tbe x direction,
h
o"1 = _--
Nx
the
1) for
web
each
is
treated
as
a layer
whose
b.
are g_ven by eq. (4. 8.4) and (4. 8. 5).
and
CZ, 3
-- 0
(4.9.2)
O
4.0.001
where
A
For
is
the
rib
cross
the
1 rib
in
the
4_
_1
= "
h
3-_"
o
sectional
area.
= bd
A O
for
unfl_nged
isogrid.
direction,
Nx
0
2,
2h
3A
3
(4.
x
9.3)
O
The
preferred
direction
for
4.9.
1
Design
Situations
T_I
Typical
uses
for
structures
where
another
good
aumber
of
built-up
design,
for
to
4.9.2
The
for
well
as
to
information
of
of
is
general
instability
andEuler
profile
are
likely.
would
in
be
important.
the
for
_
openisogrid
_ost
for
such
provide
a
regular
5 for
information
on
machining
of
baffies
pattern
will
isogrid,
thrust
would
attachment
nodal
the
to
points
geometry
including
be
reduce
compared
of
on
and
slosh
structures,
pattern
Section
direction.
interstages
Isogrid
and
assumes
column
EULER
that
minimum
buckling
COLUMN
4.9.002
j,
1 ribs
the
and
nodes.
Optimization
optimization
equally
the
The
complexity,
as
is
cylinders
application.
Refer
Method
method
isogrid
potential
parts,
7
ribs
accessibility
equipment.
Section
open
the
of
BUCKLING
weight
the
ribs
occurs
within
when
the
cylindrical
Under
these
d,
the
and
assumptions,
ratio
of
equivalent
rib
width,
The
magnitude
weight
grid
thickness,
T,
the
plate
thickness,
spacing,
b
a
are
determined.
are
determined
the
circumference
local
Euler
will
not
satisfy
and
column
_,
make
make
the
the
rib
the
triangle
rib
buckling
of
the
is
shown
that
1
E*
t.2
R
depth,
ribs
width,
size
d
in
b,
the
>
b,
radial
and
the
an
integral
the
rib
grid
spacing,
a,
multiple
width,
direction
of
of
so
that
the
cylinder
occur.
Ge ne ral
In
to
of
Instabi!.i_tv
Reference
written
2-1
in
the
Ncr(1)
it
_l-v
2)
independent
of
is
For
uniform
see
Figure
instability
due
to
bending
may
be
form,
=
wkich
general
the
compression,
4.
(4.9.4)
cylinder
F,
the
length.
cylinder
is
highly
length
dependent,
Z-l.
Let
N x
= N a
+ N b
_vhe re
Na
then
it
-
is
independent
N x
:
F
211R
'
shown
in
of
the
Na+N
b.
Nb
=
Reference
cylinder
Ivi
IIR 2
. Nb__.
'
N
a
2-1
that
the
buckling
strength
of
length
and
may
be
by
simple
= Ncr(l)
given
the
the
cylinder
formula,
(4.9.5)
4.9.003
is
t*
and
eq.
E::" from
(4.9.
(2.
eq.
5.3)
and
(2.
5.4)
may
be
substituted
into
4).
Ncr(1)
_
2Y
_E_-
1-v
=
where
Y
is
the
A
area
and
I
are
O
and
r_-
(4.9.6)
'1correlation
'_transformed'l
If
_tE
.__
2.12
and
the
factor".
moment
actual
rib
In
of
inertia.
area
and
eq.
(4.9.6)
moment
A and
of
inertia,
I are
then
the
since
A
O
I
are
linearly
proportional
A
A
-
h
to
the
width,
I
o
I
'
-
o
h
giving,
Ncrtl_,,
For
_
Rectangular
2.
12_/ERh
_--/'_o
Io
(4.9.7)
Stiffeners
A°
-
I_
Io
"
1--_-
d
L
Ncr_l
} =
0.613
N E
(4.9.8)
bd2
Rh
4.0._
For
All
Material
in
Flanges
,I
!
Ao
= ZAf
Io
=
2Af {_-/2v!
I
=A2d
Afd
A
o
From
o
2
2
2
f
(4.9.7},
N
Only
the
From
2.12NE
-
cr
Ad
f
Rh
Rectangular
(4.9.8)
Stiffeners
for
Y
Ebd
Ncr(l}
= 0.460
Define
the
(4.9.8)
Will
= 0.65
Be
Optimized
and
2
ratio,
then,
Ncr(l)
Euler
Column
Assume
From
Buckling
the
eq.
I ribs in the
F :bs
cbdirection
(4.9.3)
2h
3bd
P
bd
x
2
P
o,
= 0.460
-- "_h
a
N
x
:
_
_
N
x
4.9._
R
{_.9.9)
By the
Euler
formula,
p
kc
I12 EI
2
a
whe re
k
is
the
-
fl Z Edb 3
k
c
column
fixity.
c
Thus
N
x
-
kc I12 Edb 3
2
12a
a
k
E
db 3
c
= 1. 422
3
a
I
Ncr
(2) =
1.42Z
(4. 9. 10)
k c Ed
_3
(4.9.
101 for
|
Equating
(4.9.
9} for
general
instability
and
buckling,
Euler
column
.... ..
0.460
d
-R
=
^2
P
1 422
'
=
k
O. 323
k
-pZ
c
d
R
(4.9.
C
Equation
(4.9.
11) for
Z
O. 2i2
Nx
_
may
(0. 323)
now
be
substituted
E2 d5
kc
O. 0683
\ER]
=
kc
4.$ .006
into
eq.
(4. 9. 9)
11)
l/5
d[
-
After
d
has
been
found
into
(4.9.
11)
and
_
fromeq.
= b/a
(4. 9. 12)
14.6z kc \ER/
may
(4.9.
be
12),
d/R
can
be
substituted
determined.
Since
_" =
3 bd
h
_
p_ x_- b___d
a
(4. 9. 13)
So that
the
4.9.
Worked
3
Isogrid
equivalent
with
weight
thickness,
"t,
has
now
been
determined.
that
Euler
column
Examples
flanges
Since
this
case
is not
of the
bars
is not
critical.
Nx
=
9,000
lb/in
E
=
I0.6
(106)
R
=
198
in.
d
=
2.00
in.
h
=
10. 00 in.
=
0.65
optimized,
it is
assumed
psi
4.9.007
buckling
From
The
eq.
(4. 9. 8)
Ncr
_
2.12 i98
(0.6s)(I0.6)
(10. 00)
9000
=
1.475
(104 ) Af
Af
=
9.0075
14.
::
equivalent
t
=
weight
6Af
h
For
comparison
ence
2-11
Reference
from
(0
cr
9000
2
The
= 0. 366
with
for
stiffened
cylinders
is valid
_
values
are
the
lower),
in.
same
¥,
(actually
on:y;
for
unstiffened
usinr_
to
for
0. 328
(I05.)
t
the
(4.9.4).
=
2
-
0. 612
o
R
0. 612
(10.6)
198
9.00
= _
t
= 0. 524
ratio
is
monocoque
to
O
610)
t,
with
Et
N
in. 2
thickness,
6.0
10.0
2-6,
eq.
0 " 610
(2.00Af)
of open
=
106
to
2
=
0. 275
isogrid/monocoque
2
O
is,
¥ = 0.65,
Refer-
cylinders,
monoccque
thickness,
Unflan_e_ ;sogrid
N
(Optimized)
=
9,000
Ib/in.
E
=
10.6
(lO 6) psi
R
=
198
in.
=
0.65
eq.
(4.9.
X
From
12)
1IS
w
d
P,.
__--
Assume
k
=
c
l_.l _
=
(_)s
d
K
d
eq.
1o 6 (i0 _') (198)
=
14.6Z
_
1.40O (I0 -z)
=
1. 400
(4.9.
:
[
9.o i,o' )
\EF,/
From
Z. O0
(P) 18.35
(198)
10 -2
i
:
2
[4.28
(10"12 ) =
=
2.77
(I0-6)
537 (10 IZ)
_
in.
11).
0. 323
z.--TV6(l.
4) 10
O. 0475
b
= --
-z :
3
o. ZZ6 (I0"2)
a
4.g._
=
18.35
(I0 =Iz)
.-:5.37
(10 10)
-
Since
be
one
about
The
should
have
Take
it is evident
that
a value
of a < 40
in.
would
right.
circumference
c
d > b,
=
c,
211R
2II (198)
=
35 pockets
is,
around
the
=
1243 in.
circumference::'
Then
a
1243
= _
35
b
= _a
= 0.0475
(35.
equivalent
weight
thickness
=
35. 5 in.
and
The
-t =
3
"_
3.47
=
bd
h
-
5)
?. x_-_
(0.0475)
=
1.690
in.
<
2.77
in.
is,
d
2. 77
= 0.456
in.
m
The
relative t's are now:
n
Construction
t
O. 524
Monocoque
Unflanged
Flanged
':'If the
frame
spacing
buckling.
a,
Isogrid
0. 456 in.
O. 366 in.
Isogrid
is too large,
in.
it will be necessary
4.9.010
to check
for between
The
unflanged
grid
may
As
a check
on
eq.
(4.9.9)
and
(4.
9.
10).
be
isogrid
example
opened
the
up
sho_,s
that
the
grid
spacing
of
the
flanged
iso-
considerably.
unflanged
example,
substitute
the
derived
dimension
2.772
Ncr
(l)
=
0. 460
(10.
N cr
(2)
=
1. 422
(2.
=
8950
lb/in.
6)
0)
(10 6) (0. 0475)
(10.
6)
(close
(106)
enough)
4,9.011
(2.
--]-_j_-
77)
(0.
_-
9o00
0475)
3
lb/in.
into
4. 10
OPEN
AND
SKINNED
For
stiffened
plate
supported
triangular,
or
irregular
important
structural
rigidity
of the
region;
its
isotropic
latter
two
properties
able
solutions
typical
etc.
, in short,
design
and
depth
or
may
the
web
area
all
areas
are
Frorr_
the
centers
as
is
natural
4. 10.2
Analyses
Skinned
Isogrid
From
eq.
with
In
for
(Z. 2.4)
point
points
twisting
over
exist;
M
apply
some
a wide
and
and
the
the
IN.
These
many
avail-
2-3.
floors,
subjected
loads.
construction
for
plates
two
thin
pattern
with
mechanically
construction
lightening
fillet
insert
web.
inches
Open
material
to secure
drilled
tends
to build
equipment
or
face
reliability
holes
If
is unskin-
attached
the
of view,
stnld
to bending
the
inspection.
for
containers,
required,
isogrid
the
walls,
less
isogrid
ribs
since
is high
out
in
since
at the
up,
serve
and
transfer
plate.
and
(Z. Z 5),
: 1
--
Mxy
high
loading
resultants,
and
where
Mx}
Di:
My
the
directions
doors,
are
such
fabrication
points
be
is
an
accessible.
to the
possesses
are
the
Keference
be integral
constructed
attachment
loads
weak
stress
which
be
grid
these
to immediately
of fluids
up
constructional
of the
local
flow
built
open
no
would
may
plates
construction
theory,
surfaces
free
be
isogrid
circular,
Situation
flat
it ,-nay
rectangular,
to distribute
it possible
plate
construction
sandwich
acts
in plane
situations
all
accessibility
The
and
Design
with
Specifically,
so that
make
locally
the
character
of classical
Some
loaded
which
bending
Typical
PLATES
boundaries,
construction
of the
ned.
or
advantages.
uncoupling
4, 10. 1
ISOGRID
×
x
(4. I0. I)
Y
I-L'
-_
D (2Xxy)
(4.
4.10.001
10. 2}
Solving
for
the
change
/:/
X
=
af
curvature,
[::]
- D(1 - v 2)
_
2
(z Xxy)
-
D (1 - u)
Mxy
eq.
(2.2.3)
these
become,
Using
My
×
D (1-v
2)
-u
M
Y
2Z
_xy
Now
=
define
-Z
(2 Xxy)
=
-]3(1
- v)
the
section
modulus,
S,
M xy
by,
9
-_- D(1-v-)
Z
ES
(4.
10. 3)
(4.
10.4)
(4.
10.5)
neutral
axis
then,
x} [ li
1
1
-v
Mx
= b_
_y
Yxy
Section
The
to the
-v
::"
2 (1 + v)
ES
lj
, My
M
xy
Moduli
maximum
extreme
fiber
stresses
requires
the
distance,
Z,
from
the
t
fibers.
4.10.002
t_
1.00
I
i
TRANSFORMED SECTION
Part
Ai
_e
©
®
A
=
bd
h
ZA.
=
i
bd
t + -h
=
The
neutral
o
ZA.
fiber
distance,
c 1
The
at
ZA.C.
= ,, I _
Z
The
axis
section
=
-
(t
a distance
a6-
2
Cl,
+ Z o)
modulus,
+ a
the
skin
=_(t
_
SI,
of
-t2/2
d/2
bd 2 / 2h
2a+
the
(4. 10.6)
from
_
=
o
from
the
neutral
is
given
by,
axis
is,
I
"1
of
-t/2
-
Zo
= t
1
Ai_i
t (I +_)
A :
bd 2
A% = z Ai_i = _-
1
1a + 5 _+
skin
is
1)
given
(4.
by
(4.
I0.7),
(4.
I0.
3)
10.7)
and
(2.3.5).
1
t2
6
2c,+
132
c_ 6 + 1
(4. 10.8)
4.10.003
The
fiher
c2
The
distance,
=
the
t2
---6
that
skin
factor
Thas
the
Skin
d - Zo
ribs
Z6
from
the
neutral
axis
is,
+l.a
a 6 + 1
(4. I0.9)
Sz, of the ribs is given by (4.10.7), (4.10.3), and (2.3.5).
_Z
Z 6 +a6
the
first
without
due
of the
: t_._
2
section modulus,
S2
Note
e 2,
factor
ribs
and
rib
grid.
to the
for
monocoque
_--I
, a--O
familiar
value.
+ I
(4.10.10)
of (4. 10.8)
and
the
factor
is a nondimensional
=
tz
= -_
second
(4. 10.10)
is the
section
modulus
of
amplification
construction,
,
5 --0
for
the
and S 1
- S2
Stresses
From
Hooke's
law
skin
{2.4.6)
(4.
10. 1 1)
(4.
10. lZ)
(4.
10. 1 3}
1 -v 2
O'y
E
"rxy
Substituting
-
2(1
+v)
(4. 10.4)
_xy
and
(4.
10. 5) into
this
= -S1
(T
4.10.004
relation,
M
xy
Rib
(4.
S 1
10. 14)
Stresses
Because
of the
uniaxial
character
of the
ribs,
the
rib
stresses
are
given
by,
_.
=
E e.
1
where
1
e i is obtained
from
E
4
_r_
(4.
3
10.4),
(4.
Vxy
r _
o-3
From
1. Z)
00]
o_1
o-2
(2.
X
10. 15)
/
(4. 10.5)
{X}I10 °j
-L,]
Yxy
-
ES
0
Z( 1+ v)
-v
O
10. 16) ir'o
(4. !0.
+y
Substituting
(4.
o"1
1
(4. 10. 16)
M_y
My )
15),
"4
0
(l-3v)
Z_(I+v)
-4v
1
_Z
_3
(3-v)
4S z
(l-3v)
-Z _
_l+v)
4.10.006
(3-v)
Mxy
Mx
M
Y
and
since
u = 1/3,
m
o-2
l
s2
,
i
1
0
2
o
2V- _
$
2V_
2
0--_
o.3
.1
Q_
3
M
m
Section
To
Modulus
facilitate
(4. 10. 17)
xy
Mx
M
Y
Graph
the
calculation,
the
section
moduli
are
written
in the
form,
t2
S1
-
k
=
1
2
k
k
6
20
{32
+ab
4. !0.
+ 1
t2
= _
k2
(4.
2
a2
is the
section
kI
.--. k l (a,
k2
=
10. 20)
are
the
rib amplification
by
the
modulus
of the
skin
alone
and
6)
k 2 (r_, 5 )
graphs
19)
4. 10.21)
25,o6+1
t 2 /6
where
4. 10. 18)
1
in Figures
factors
in non-dimensional
4. 10-1
anti
4. 10-2.
the
ribB
When
a --..0 but
6 = constantS0,
width
becomes
very
have
small.
4.10.006
form
constant
depth
which
are
given
but
the
rib
:..+ i+.'_
. ; ,::++
b.-i--++_
::[
:
L
:::;::::
:;:';+';;
!!!17_?! e
''
t-,t+
::;+:;:
.....:
liIJ
I+
,++1,.+_
:_:!:':+{
.+++..++
.,+..R..++.+.+.+.+-_.
+,+,,_-.I+
_
!?i
-
:;:t_++;:
':![!:!
....
....
_
Ii,+_
+t++,
+- .it++,
+., ,+++-+,
+.+++++.
+..H.+
i++i14_+
...,+++
t.+++,+,
!!iiit:
.
It+_+,
+4
+++."
II
+i+
I
i::11:I11:
v
e,,-
4.10.007
CR 169
25
F- .....
3O
t
I
28
26
24
22
2O
18
.....
|
16
k2
14
12
10
03
Figtjre
4.10.2.
k2,
o,
8 Curves
4.10.000
04
In
such
cases,
attached
stresses
to
the
pathological
be
Open
Isogrid
this
limiti
n,a case,
Froa_
eq.
(4.
ed,,e
weakenthe
where
the
skin
eq.
(4.
10.
13)
and
now
given
by
(4.
10.
17}
where
(2.
3. 2).
and
in
critical
3 1--!(t.dlt
thin
the
stress
10)
occur
a very
stresswise.
for
10.
graph
of
structure,
tile
avoided
are
outstandin,_
on
case,
equations
the
actually
cases
should
In
skin
at
domain
deep
rib
These
k <1.00
and
structure.
,
(t : -TT_ (_
dt
_ bd/
h
(4.
10.
14)
S 2 is
_iven
, _-
}
drop
The
out.
by
(4.
10
rib
I O) from
the
l
)
:
-7-
m
S,
t2
S
Z
1
bd 2
(,
h
(4.
10.Z2)
(4.
1o.2
and
gl
" gz
4.10.009
_
4.
10.
3
Worked
E,:amp!es
Skinned
is_grid
Simply
supported
Circular
plate
R
:
36in.
q
-
3psi
Ftu
-- 60
From
ksi
Reference
M
2-3,
:,)vl
_
x
_
808
Assume
From
the
k
(3)
lb
maximum
stresses
at
the
center
2
qR
362
in
b :
10,
(_-
kl,
k 2 graphs,
0.20
-- 28.8
1
k2
+v
16
y
3.333
16
the
:
4.25
2
S 1
t
:- --_-
k,,
S 2
:
t2
"_-
k2
-
on
tension
:
_3
Assume
ribs
_1
tz
'r2
_28.8
4.25
--'_--
t2
t
2
0.710
60
t2
Then,
from
eq.
808
0.
su8)
3 (0. 710)
:
side.
2
T
t2
- 4,81
710t
2
60
000
.- ! zs7 (1o "z)
000-
4.:0,010
(4.
10.
17)
of
the
plate
are,
t
=
d
O. ll3
in
6t
10
5
a
b
(0.
-
113)
10
0.20
:
h
--
b
::
d
1--0
b
:
O. 113
in.
h
:
50
(0.
113)
:
5.65
grid
size,
a.
is
given
by,
1. 13
in.
b = 50b
Let
The
Then
h,
Oo 866
a
As
say.
a check
on
:
5.65
--:
O. _¢,6
the
_
skin
o"
cr
ffC
The
r
:
:=
skin
12(1-
=
(0.
,
2
_5, 000
psi
wright
(1.60;
>
from
(4,
= -13,
150
13)
psi.
Rcfer-,ncc
t
l,.,0E (=)
(0.
113_ g
) \6.53
l
1 {, 500
thickm,_s,
:
10.
2
stress,
(-;/2
{10.7)(106
0.113
113)
buckling
11 ' 10
equivalent
in
808
4.81
allowable
6.53
stresses,
-
fix
The
in.
1-,
0.181
is.
in.
4.10.011
2
2
9,
is
4.
11
MINIMUM
TO AXIAL
In
general,
the
length
the
variation
the
of
constant,
ing
the
plish
OVERALL
WEIGHT
FOR
CYLINDER
COMPRESSION
AND
BENDING
envelope
the
cylinder.
of
load,
for
the
for
h
select
N
held
cr
(i)
Ncr
Although
b,
and
nmst
rapidly
and
stations
between
the
_oint.
and
constant
along
points
determine
the
t may
h,
problem
the
as
well
will
be
varied
to
reflect
usually
be
held
must
becomes
length
along
the
tank
the
associated
of
that
of
minimiz-
tank.
To
accom-
where
Ncr
the
axis
distance,
Ai
Ncr
LOAD
i,
for
a set
the
value
given
is
at
mid-
_
(t + 1)
TANK
construct
along
f_
Nc r
Next
vary
Pburst
height,
reasons
i
and
d
triangle
at
changing
N cr
maximum
grid
fabrication
weight
this,
of
SUBJECTED
of
graphs
of
Ncr
of
_
AXIS
ENVELOPE
vs
h
as
functions
of
p
for
each
{i).
i
h
R GIVEN
]
I
I
I
t'/h GRAPH
4.11.001
""
station,
From
the
relative
set
tank
of
_
t,
h
weight
graphs
may
be
for
. th
:
the
the
determined
station
for
along
various
the
tank
value
of
axis,
h.
"T¢h}
=
n
Ai
i-I
_m_n
I
v
hG
h
hMl N
._
T (h) GRAPH
The
initial
of
h
tank
value
which
axis.
be
h
remain
penalty
satisfied
Since
minimum
may
of
the
weight
allowed
the
ho,
called
all
the
graph
burst
-of
t
must
in the
each
station)
is
with
only
minor
changes
over
the
be
conditions
at
constant
on
h,
entire
quite
at
the
flat,
does
as
n
neighborhood
in
tank
selected
so
of
not
that
impose
smallest
stations
considerable
_
the
to
(the
alon_
value
variation
the
value
the
for
:n
h
requirement
that
a significant
weight
design.
•
4.11.002
I
i
4.
12
NOTE
Although
tion,
by
ON
USE
OF
the
a,
6
curves
eye,
since
the
variation
eye
along
the
interpolate
by
appvo×imatety
It
will
5 ::
30
x,
ential.
distance
found
that
at
the
extreme
scales
almost
coincide
of
_raph
betwecn
the
A procedure
A.
which
Locate
of
B.
C_
o,
6
the
paper
so
If the
computed
cannot
direction
of
the
graph
are
to
the
_,
_
point
and
right
of
the
the
The
edge
the
paper
the
_rid
left
may
now
superimposed
may
now
is
not
followed,
otherwise
be
read
with
be
with
sufficient
then
readily
the
_
5
to
is
=
Z or
Moreo_'er,
the
bottom
as
follows:
up
a small
piece
graph
curve
and
direction.
also
mark
the
scale
wit)-,
point.
to
sca!e
calculated
precision,
4.12.001
_
interpolated
checked
initial
the
transferred
upon
a
agree
be
curves
move
at
to
variation
is
convenient
along
a
easy
= 3.0.
aligned
of
the
at
is
values
not
scales
edge
_rid
rarely
x
and
paper
wil!
x
the
pencil
the
bd/th
and
_, direc-
identical.
by
of
procedure
practically
of
most
the
the
is
because
point
edge
above
be
is
it
along
portion
:< = 0.50
the
intermediate
This
the
along
linear,
a
its
values
interpolate
approximately
the
Mark
of
to
in
found
that
easy
direction.
values
be
CURVES
a'
with
wilI
,5
fairly
is
edges
these
the
_,
are
expon
be
y;
values
the
x
The
values.
x
scales.
values
of
a
since
the
graph
on
the
frmn
4. 13
OFF-OPTIMUM
For
many
of
machining
applications,
by
than
larger
4.
than
13.
the
be
other
than
that
the
plate
thickness
gridwork
basic
less
than
it
follows
that
consequently
not
critical
are
general
instability
condi
ions
assumed
the
circular
cost
design
required
that
availability,
triangle
assumptions
tt_at
be
size
underlying
be
the
analysis.
to[a!
plate
thickness
than
the
General
thickness
the
the
Analysis
optimum
critical
plate
are
of
than
is
and
These
to
gridwork
it
thickness
optimum.
greater
the
Usually
Method
thicker
is
optimum,
and
requirements
the
optimum
and
the
rib
the
grid
dimensions
for
rib
and
skin
triangle
will
be
cripphng.
freckling.
Instability
The
critical
external
and
optimum.
optimum
tbc
due
that
1
is
The
require
off-optimum
When
size
the
the
following
constraints
etc.,
indicated
less
!SOGRID
loading
pressure
or
bending.
(Reference
are
cylinder
Subsecticms
4.
either
under
1 and
4.2.)
the
sph,,-rical
combined
axial
cap
under
compression
Thus,
)
E
cr
whe
'
t '_
o
R
P
R
(4.
13.
1)
(4
1 _. 21
14
I _. _)
re
N
(L _t !'
cr
CO
for
the
0.2h0
spherical
(lb)
N
cr
cap.
:
N
x
C (1
F
M
Z "R
-1t _"
)
O. 1397
4.13.001
¢
for
the
circular
Skin
Buckling
For
skin
N
buckling,
(2a)
cr
C
for
cylinder.
:
C
-
3.47
l
the
spherical
C 1
=
the
circular
1
(4. 13.4)
and,
cap,
(4. 13. 5)
for
10.2
Collecting
cylinder.
requirements,
Z
N
(I_ = c
cr
'
N
i2) ---C
cr
Given:
S
= t+d
o
(4. 13o6)
E--!-tO
R
t2
Et(l+a)---_I
h z-
(4. 13.7)
(4.1 3.8)
_< S
o
Given:
h
(4. 13o 9)
._ h O
Equations
(4.
geometric
dimensions,
remaining
three
From
(4.
S
13.6)to(4.
dimensions
13.9)
consist
b,
t
d,
are
and
of
h.
four
Of
determined
equations
for
these,
h
is
as
follows:
already
the
four
required
given.
The
1 3.8)
:: t*d
:: t(l+6)
t
14.
:
4.13.002
1 3.
10)
Substituting
14,
N
13.
10)
into
E
R
s2_2
(1+6)
= C
cr
N
o
R
cr
13.6),
_3
(4.
13o 11)
(4.
13.
12)
3
(4.
13.
13
s 2
(4.
13.
14
(4.
13.
15
E
s
2
(I+6)
14.
1 3. 10)
and
(4.
C
E
C
(4.
o
From
1 3.7),
3
N
N
--
cr
1
s
(1+o_)
h2
(I+6)3
h2
l+a,
cr
C
E
1
11+6)
s
3
Define,
N
h 2
cr
C
E
1
N
s
R
cr
C
E
O
then
x
l+a
--
(1+6)
v
::
•
3
13
(4
----
2
13.16
•
(1+6)
This
in
defines
Figure
geometric
a
4.
mapping
13-1
and
does
on
of
the
a log
plot.
This
depend
upon
not
or,
6
domain
into
mapping,
the
values
4.13.003
t
the
it
used
x,
y
domain
and
is
may
be
noted,
is
purely
C o and
C 1"
for
given
2
2.5
3
8
10 -4
9
4
3O
6
I"
25
2O
I
l
i
I
,
r
I0
!
9
I
8
16
I.
7
6
I
Y
5
X 10-2
4
I
!
1
1
2
5
6
7
8
-4
10
lrOI'D°u'r
4
l
I
,i
r_
NORMAL PR|
SPHERICAL
Co
=
0.2I
C,1
-
3.47!
c_
. o._
I
S-d+_<:'31
h>h
OPTI_
Y
X _0"2
'
"T
d
i
N¢I'
AS A CH|(
.I
i0 -2
l_qLDOU l'FRAME
IIL-I
i
I
CR169
5
6
!
7
8
9
10_
_
30
_ t
I
SUMMARY
OF OFF.OPTIMUM
EQUATIONS
t
;-_t"
t15
NORMAL
PRESSURE
SPHERICAL
CAP
:I_
!
•
I
--.
I
I.. _ 10
L>_,-: : : •
.--_.9
REFERENCE
C0
-
0.260
0.307
p. 4.13,001
C1
-
3.47
10.2
p. 4.13,002
C2
=
0.634
0.616
P. 4.13`008
S = d + t _S
:
CYLINDER
UNDER
AXIAL
COMFRESSION
AND BLENDING
ON
OPTIMUM
-::|
....
+
h >_h OPTIMUM
_._>!?
Y
X 10.2
i=:zl
--
X
-
Ncr h2
.......
Eq. 4.13.13
C1 ES3
--'.'-'4
I
_-.:
:-T-:
5
y
.....
......
=
Ncr R
Eq. 4.13.14
S
I
i
Eq.4.13.10
1+8
...._
..... 4
..........
d
"
8t
p. 4.13.006
b
=
th
(Z ---_-
p.4.13.006
i
.....
.........
+
i
AS A CHECK,
• :::::
.
Ncr (RIB CRIPPLING)
-
C2 Et (1 +41)
Net
...f
--_--
9
1,S
10-2
Figure
4.13.004
4 13-1.
X,
Y,
{n, _1Curves
The
design
procedure
A.
Calculate
13.
Read
is
x
off
as
and
a
and
interpolation
follows:
y
6
is
from
eq.
from
the
required
(4.
a,
Calculate
t
from
equation
D.
Calculate
d
from
d
E.
Calculate
the
rib
width
from
been
determined.
all
dimensions
crippling
from
have
the
13}
6
graph.
13.
10}
and
(4.
13.
14).
(Logarithmic
here.)
C.
Thus
13.
:
(4.
t
6t
a,
b
,: --_
1+5
: a th
d
As
a check,
determine
rib
equation,
2
N cr (3)
whe
=
C 2 E
{l+a){-_)
-- 0o634
for
the
for
the
(40
13.
re
C 2
spherical
cap
C 2
=
and,
0.616
circular
cylinder°
4.
13.2
Worked
Spherical
cap
Example
with
the
Per
=
33.6
R
=
198
Ftu
-
71.2
ksi,
11.6
( 106)psi
E
An
t
optimum
holding
determined.
the
reqmrements:
psi
in
design
plate
following
will
thickness
first
be
constant
obtained
while
4.13.005
and
the
opening
weight
up
penalty
the
grid
involved
will
be
in
17)
Pcr
"E"
From
--
the
Po
33.6
TT_
(10-6)
graph
in Figure
(103 )
--
=
Z.90
4.1-1.
train
, --R--
1.48
(10 .6 )
0. 00107
=
Ft u
t hu s
Po
=
1.48
t"
=
0.00107
m
N
(71.2)
=
105.3
(189)=
Pcr
Ftu
E
p
psi
0.202in.
2.90
(10 "6)
1.96
(I0 "3)
103
_
=
I. 48 (I0"3i
x
=
(103)
1.482
Y
=
_
130
the
a,
8 graph
=
0.29,
From
o
_
1.96
1.323
1.96
(10 -3)
0. 130
Ftu
p
5
=
in Figure
=
10.2
4.1-2.
14.6
the n,
1.48 (10 "3)
pR
t
=
2Ftu(l+a
d
=
6t
b
)
=
14.6
=
N
d
=
0.0888
in.
=
189
-
2 (1.28)
(0. 1092)
=
1.594
: _./1.96 0.634(10 "3)
in.
(1.594)
4.13.001
L
f
lh
J
'
0. 1092
in.
h
as
=
_
t
=
4.59
in.
= _
1.96 3.47(10 -3)
(0. 1092)
a check,
o
_
bd
th
-
t(1
÷ 3e)
t-=
As
a check
0. 0888 (1. 594)
0.1092 (4.59)
from
S
=
N
cr
y
=
the
0. 109Z
1.594
Pcr R
2
33.6
_
0.Z8Z
(1.846)
off-optimum
d , t =
=
0.202
Figure
curve,
+ C. 1092
(189)
=
in.
=
1.703
4. 13-1.
in.
31801b/in.
2
NcrK
0. Z60ES z
__
=
-
hz
Ncr
0. 0678
3.18 (I. 892 105
0. Z60 (II.6) (I0v) (1.703) z
3.18
(103 ) (4.59) 2
=
=
0.337
(10-3 )
X
3.47 ES _
From
=
3.47
(II.6_ (106 ) (1.703) 3
the o, 5 curve,
a
=
0.28,
6
-
14.7
the
previous
This
checks
with
For
the
off-optimum
first
S
:
1.703
S
=
SO
in.
values
using
design,
h
=
10.0
h >
ho
in.
4.15.007
the
graph
from
Subsection
4.1.
then,
y
--
0. 0687
as
before,
and
x -- 0.337 (10-31(1°'°_ 2
\4-:-_/
From
,_
the
a,
6 graph,
=
0.085,
6
s
1 + 6
t
Figure
_-
4.13-1.
7.8
1. 703
_
=
: 1.6o (lO'31
=
0.1935
in.
¢
d
--
6t
=
b
=
_-- th
d
7.8
=
O. 109
(0.
1935)
0.085
=
1.510
in.
[0.1935 1.Sl0 110,
0)]
in.
m
t
As
=
t(l
=
O. 243
a check
From
the
+ 3a)
on
=-
0.1935
strength,
Figure
0°085,
6
A'-
N
cr
(l)--
(I.Z55)
in.
_3 curve,
0
=
CoE
2-I
=
7,8,
_
"
5,
2
2
_
- 0.260 111.611_061Io. _93st2 (5.21
=
3110
=
ClEt{l
lblin.
t2
Ncr(Z)
+ °)h- _
4.13.00e
I
=
3.47
(11
6) (106 ) (1.085)
(0.1935[
3
(10. O) "_
•
= 3160
lb/in.
b2
Ncr(3)--
c 2 Et
(I + a) 7
= 0.634
(11.6)
-- 8050
Note
that
the
For
the
s
lb/in.
crippling
allowable,
seco_ld
off-optimum
design,
1.703
in.
:
rib
<,08,,0
,93, 2
(10
Ncr(3)
,
s = s
h
= 15.0
h
>h
O
y
:
x
0.0687,
0.337
the
a,
a
in.
o
:
From
is not
as
before,
-
.
(10
3)
and
5 curves,
= 3.60
Figure
= 0.05,
5
t
-
s
1+6
-
1. 703
6.6"
d
=
bt
=
5.6
b
th
= a-- d- =
0.05
-
in.
O. 1340
t" = t(l
2
15.0
_
+ 3a)
:
4. 13-1.
= 5.6
0.258
(0.258)
(0.258)
= 0.258
"110=3)
in.
=
1.445
in.
(15.0)
1.445
(1.15)
=
0.296in.
4.15.00e
critical.
Summary
of
s
=
d
results
+
t
=
1.703
in.
h
b
d
t
a
5
T
4.59
0.0888
1.594
0.1092
0.282
14.6
0.202
10.0
0.109
1.510
0.1935
0.085
7.8
0.243
15.0
0.1340
1.445
0.258
0.05
5.6
0.296
'::
*Optimum
design
Note
b
that
increases
become
non-critical.
4.
Summary
13.3
Normal
while
of
Pressure
Off-Optimum
on
Spherical
Cap
c
-- 0.
d
decreases.
This
Cylinder
Compression
260
s
:
d
h
>
Axial
Bending
0.616
+ t
h
why
the
ribs
Reference
Eq.
4. 13.2,
4.13.3
Eq.
4. 13.4,
4.13.5
p.
4.
13.005
3
Eq.
4.
13.
13
2
Eq.
4.
13.
14
Eq.
4.
13.10
p.
4.
13.005
-< S
optimum
optimum
N
h2
cr
-
Under
and
10.2
0.634
X
reason
0.397
3.47
:
c 2
the
Ec_uations
O
Cl=
is
C
Es
1
N
R
cr
Y
= "
c
E
s
+
b
O
S
t
-
1
d
=
6t
4.13.010
_a*
_
,4 Ip_'¸_
_.d
¸
i
th
d
As
a
p.
check,
_> N
Ncr
(Rib
Crippling}
:
C 2
Et(l_a)(--_-)
2
4.13.011
cr
4.
1_.00_
Section
The
point
azhere
the
manufacture
milling
into
a hole
ribs
of
isogrid,
extra
ribs.
The
the
center
attachment
for
of
other
The
nodal
region
in
are
flexible
and
cause
their
sheet
so
of
the
of
able
being
structures
the
may
example,
in
and
be
an
current
very
area
its
These
node
holes
fittings
practice
the
effect
are
given
reduced
by
ideal
isogrid
is
based
can
be
analyzed
as
which
accurate
in
have
"smeared-out"
analysis
on
the
stress
distribution.
in
the
detail
design
a considerable
loads.
3.
the
the
area.
on
"smearing
a solid
been
developed
for
has
sizing
the
These
amount
of
of
flexi-
local
areas.
cyclic
the
for
the
ignores
critical
out"
continuous
method
of
nodes
As
This
information
drilling
of
because
of
equations
points
Section
stresses
properties.
the
concentrated
in
In
because
cutting
considerati3n
elastic
5-1.
without
is
of
analysis
providing
experiencing
are
Figure
nodes
carrying
special
the
the
material
structure
use
at
extra
redistribution
the
important
left
this
for
appropriate
to
is
a node,
intersection
deserves
4,
called
of the
or
However,
nodes
stresses
node.
while
structure.
of
of
stiffened
with
monocoque
bility
2 and
that
penalty
a local
is
material
center
isogrid
material
advantage
isogrid
in
GEOMETRY
the
each
use
in Sections
ribs
to
structures
of
the
cut
weight
Examples
stated
intersect
isogrid
cannot
in
L
the
cutters
the
NODA
5
For
loading,
CFI169
Figure 5-1. Isogrid Node
§.0.001
the stress concentrations
initiating and growing
should be determined
cracks
which
to assess
could adversely
the possibility of
affect the st,rx'ice life
of the structure.
There
the
are
at
local
least
stress
element
two
distribution
analysis
There
are
methods
may
methods
and
excellent
in the
literat_tre,
to for
in the
another
many
refer
which
nodal
e.g.,
be
5-1
and
5-2,
5-3
describes
on the
element
An
application
for
the
I-Z
photoelastic
,nalysis
can
be
The
of stress
redistribution
amount
function
each
of the
design.
illustrate
the
method
Two
identical
Both
panels
stiffening
geometry
of the
However,
the
effect
used
was
loaded
and
the
photoelastic
analysis
values
then
was
0.075
top
that
inch
was
on-centers
of the
thick.
0. 300
with
ribs
at
the
mately
for
less.
2. 1 times
a one-to-two
used
for
geometry.
The
analysis
tiysol
4290
epoxy
re:;in.
in the
direction
particular
0 degree
ribs
were
stresses
and
skin
a
will
Reference
skin
the
course,
5-5
analysis,
0. _75
uniaxial
oi
Reference
with
was
For
must
from
the
node
biaxial
determined
to determine
of 0.75-inch
analysis
be
to the
hole
"naonocoque"
_0 percent
90 degree
a node
than
and
one
and
the
pattern
tension,
wide
configuration,
A description
photoelasticity.
inch
this
analysis
of
in uniaxial
The
to isogrid
is,
one
a
based
flexibility
from
compared
analysis
to node
fabricated
was
reader
NASTRAN,
vehicle.
were
other
the
5-4.
example
for
analysis
which
structural
launch
stiffening
three-dimensional
were
were
due
element
of NASTRAN
Delta
following
panels
for
in Refe,'ence
of flexibility
isogrid
rib
found
code
a finite
photoelasticity.
References
computer
in Reference
is to use
finite
Reference
detern_ine
method
on
general-purpose
is described
One
references
information.
method.
to accurately
three-dimensional
widely-used
finite
used
area.
is to use
general
basic
can
stress
loading,
0. 162
inch
inch
thick.
in the
panels
and
The
panel
skin
5-5.
thick
with
The
a flange
nodes
at
were
milling
cutter
and
was
inch
at
flanges.
were
approxit_ately
0._a8
the
the
S0 percent
radius
F'or
greater
stresses
were
a stress
concentration
of approxi-
node.
Using
superpositi,m
concentration
at
loading,
occurred
stress
6.0.002
at
the
approxin_ately
the
the
5-inch
The
rib
the
stresses
0.045
0 degree
Three-dimensional
dian_eter.
inch
the
rib.
of the
node
was
I.
3 times
the
skin
biaxial
loading,
a
stress
in
the
stress.
stress
skin
away
and
60=degree
using
superposition
for
a one-to-one
concentratio-
of
approximately
1.2
times
from
the
0=degree
for
panels
with
a different
the
nodes
of
test
the
Finally,
the
ribs
ribs.
occurred
Other
geometry.
panels
values
For
described
in
the
skin
directly
beneath
would,
of
course,
be
example,
above,
the
adding
could
found
material
significantly
at
alter
the
results.
In
addition
to
the
of
isogrid
stiffnesses
with
very
can
be
to
flexible
made
tension
load
verify
the
bending
property
is
of
the
a
5-7,
figuration
in
across
the
nodes,
distributions,
e.g.,
and
to
very
economically
verify
the
stiffness.
structure
stress
Reference
stress
configurations
simply
a
as
local
and
is
distribution.
showed
which
the
should
be
large
center
using
extensional
a
is,
not
as
sensitive
Buckling
tests
diameter
bi-ndlng
of
agreement
was
node.
0.0.003
by
holes.
Lexan
course,
to
of
to
test
for
This
determination
designs
panel
subjected
a bending
load
essentially
local
isogrid
theory
of
to
a gross
concentration
Lexan
percent
extensional
plastic
and
v,"ith
60
and
verified
stiffness
Stiffness
excellent
hole
the
effects
cylinders,
for
an
the
total
isogrid
distance
con=
Section
6
TESTING
i
In
order
to
verify
as.
fabrication
variables
components
and
tests,
in
nature
which
other
be
Lexan
may
6.
is
small
virtually
model
tests
scale,
subscale
very
sensitive
to
by
model
tests
of
random
to
assess
necessary
in
test
or
representative
size.
a
random
testing.
or
variables.
small,
Buckling
of
full-scale
chloride,
that
of
variables
polyvinyl
so
effect
full-scale
fabrication
fabrication
check
the
to
or
subscale
non-destructive
that
For
effects
verification
example,
reinforcement
On
Lexan
the
plastic
Moreover,
relatively
such
cheap
multiple
;_odels
out
interaction
and
are
generally
insignificant
for
models,
of
theoretical
assumptions
be
made
are
all
to
be
of
fabrication
same
order
as
the
Instrumentation
effect
of
of
the
pressure
effects
low
to
to
elabo,'ate
elastic
may
be
would
be
on
loading
the
affects.
best
are
of
the
entirely
scale
the
on
order
only
practical
masked
by
metal
specimens
connodes,
gradient
conducted
it
by
elastic
around
thermal
effects
larger
the
models
are
models
of
the
readouts
Because
which
typical
of
buckling
such
data
effects
which
where
geometry
concentration
influence
tests
investigated,
objective
deflection
of
cases,
nodal
stress
and
examples
some
of
holes,
verification
In
obtaining
effects
around
curve
models.
mode
is
small
variable
tests.
are
and
TESTS
interaction
cent
theory
it
assessed
tested
recommended
values,
the
design,
free
are
fabrication
stants,
be
scale
repeatedly
model
in
only
of
the
are
can
MODEL
Since
shells
made
be
1
on
particular,
hand,
can
umptions
on
buckling
small
scale
of
l0
or
1 5 per-
means
the
of
random
which
is
of
simple
load
the
investigated.
may
vary
three
widely
ranging
dimensional
moduli
of
polyviny[
obtained
by
pulling
1.
6.0.001
photoelastic
chloride
a
from
vacuum
and
on
the
investigations.
Lexan,
interior
internal
of
the
Temperature
control
important
and
features
present
problems
accounted
for.
accurate
determination
of
elastic
model
testing.
The
of
glued
softening
the
moduli
of
the
the
models
is
also
important.
to
obtain
the
correct
of
in
an
influence
material
moduli
are
joints
and
may
should
be
z 1
//
/
J
j/
Thejpr_oper
.-'_vith
scaling
D
uniform
and
extensional
model
In
it
is
necessary
stiffness,
K.
may
buckling
depends
for
walls,
monocoque
t ':_.
factor
of
upon
linearly.
the
Thus
be
it
and
the
shown
in
for
isogrid
substituted
cylinders
R/t
As
may
L/R
proper
be
ratios
in
modeling
R
t
monocoq,le
L
R
monocoque
L
= _-
monocoque
-
shown
Section
that
For
bending
Z,
an
using
the
the
buckling
a non-linear
models
stiffness,
equivalent
equivalent
E':"
and
strength
manner
and
upon
values
by
analysis
E
is
R
t*
and
N
N
cr
cr
E
The
N
/E
effects
may
E ':"
thus
be
ratioed
from
the
built
into.the
test
cr
while
the
proper
Rib
crippling
R/t
andSskin
equivalent
monocoque
instability
characte
6.2
SUB-SCALE
Since
sub-scale
to
examine
the
test
account
here
setup
L/R
which
of
course,
of
value,
are
model,
cannot
be
obtained
primarily,
from
for
general
is
important
ristics.
FULL-SCALE
full-scale
aspects
of
typical
be
effects,
models
and
on
must
buckling
AND
all
of
for
and
the
tests
the
can
influence
expected
problems
TESTS
to
be
of
very
expensive,
fabrication
test
results.
Reference
the
expected.
Some
convenience.
6.0.002
it
and
of
peculiarities
2-1Z
gives
these
are
of
a good
repeated
A.
Examine
and
the
specimen
skins.
Straighten
defects.
Make
sure
atmospheric
Check
C.
Conduct
analysis
various
modes
specimen
can
E.
and
patch
ribs
specimen
result
Monitor
roundness
to
assess
failure.
in
rib
by
the
fatigue
bearing
gages
bent
the
over"
that
Check
bent
for
of
"bridged
provided
D.
for
is
and
undersize
ribs
understrength
protected
from
corrosion°
B.
be
carefully
is
fit
between
large
local
crippling
located
on
and
straightness.
the
effects
of
Very
local
high
not
of
in
skin
edges
of
loading
the
buckling
outstanding
may
of
isogrid
consideration.
and
overloads
the
regions
redundancy
important
specimen
and
en
undersize
degree
an
tolerance
head.
"ttigh
spots"
specimen°
by
ribs
back-to-back
and
strain
centers
of
skin
panels.
Back-to-back
so
that
skin
gage
divergence
may
be
directly
rc_douts
on
of
indicating
gages
monitoring
equipment
local
change'
should
of
be
ct:rvature
read
or
together
ribs
and
identified.
I
STRAIN
LOADING
I1=,,=
In
specimens
general
are
which
instability,
invaluable
excellent
in
BE
D
NO
NG
are
weaker
in
such
gage
readings,
predicting
not
of
ultimate
indication
I
I.j_
the
only
t l
rib
BENDII_IG
crippling
rib
load
and
skin
properly
located
crippling
and
to
I
6.0.003
be
expected.
skin
buckling
than
on
specimen,
the
buckling
in
but
are
Section
7
MANUFACTURING
7.
1
INTRODUCTION
To
date,
with
waffle
open
walls
frames,
and
and
usually
constructed
curved
spun
or
7.2
ribs.
The
shallow
milled.
In
for
testing
was
sculptured
In
very
of
in the
woald
same
ribs
compound
a
case,
under
NASA
contract
forming
built-up
the
on
known
wings
flat
The
one
- fiat
and
of
therefore,
shapes
shallo,_
Workshop
flat.
with
surfaces
is
machined
alter
general,
curved
curved
for
parts
and
s.abseq'aently
S[<ylab,
since
apply
to
curved
small
it
beams,
parts
spherical
are
segment
NASS-11542,
with
a compound
a template-guided
integral
structure
structure
has
elements,
cylinflers,
and
surfaces
control
used
han_J
deep
been
enough
to
applied
to
simple
ann
the
cones
in
aircraft.
MACHINING
the
aerospace
machining.
but
Orbital
competitive
singly
slightly
in the
left
router.
structurally
been
was
par£
it
has
floors,
wing
dome
fiat
grid
chem._.cally
operated
unless
curvature,
The
forms
In
construction,
compoand
formed.
be
TECHNIQUES
industry,
Intricate
the
mechanical
Forging
properties
deliver
the
integral
shapes
tolerances
effective
structure.
This
elements
in
enough
structural
require
finish
importance
sizes
design
best
necessary
for
achievement
legitimize
Machining
of
integral
the
limitations
structures,
construction.
7.0.001
ribs
for
word
pro_'ess
required
for
large
and
"integral".
especially
does
weight
structural
economy
are,
casting,
material.
the
very
of
by
wrought
but
stiffening
true
to
with
obtainable,
particularly
means
produced
competitive
is
features
the
thin
generally
economically
not
the
and
machining.
in
most
are
usually
close
the
are
properties
are
construction
combining
Forgings
therefore,
so
of
for
prime
waffle
not
Isogrid,
like
all
o_her
of
_he
inside
contour
See
Figure
7.2-1.
is
used
wherein
are
the
two
the
basic
geometric
types
of
waffle,
the
pocket
and
When
the
stiffening
cutter
extends
variations
so
latitude
to
cover
L._ pocket
clea_ls
out
ribs
far
it
range
the
material
in
flanoed,
a
fashioned
shank,
employed;
a wide
An
all
are
beyond
milled.
Nigure
they
of
end
design
mill
traces
the
center.
cutter
i. 2-2.
These
have
providod
ad_q'-_ate
load
intensities
and
local
reinforcement.
Six
abutting
tria_agular
dimensio:as,
define
small
and
carefl,
l attention
m tst
nodes
to
weight,
to
make
For
th,"
the
waffle
red.tee
the
adeq-late
pocket,
the
obtain
close
ho!e,
pockets,
with
corners
geometry
of
a waffle
nodes
large
be
severe
given
it
to
e.g.,milling,
cutter
larger
Lhe
the
ct,tter
toleran_es
be
noted
or
deeper
diamater
the
penalties
that
other
a cutter
m_lst
the
by
l_ockets
result,
Holes
d_scharge
reasonably
rounded
When
geometry.
electrical
rigidity,
node.
weight
:aodal
should
appropriately
cutter
are
l'herefore
are
drilled
methods
m the
may
be
used
cut
a waffle
machining.
penetrates
to
be,
Rigidity
is
smooth
sqrface
finishes
required
to
which
CRICC
CUTTER
PATH DURING
FINAL
POCKET SIZING
CUTTER
CROSS
SECTION
(RADIUS
SMALLER
THAN
CORNER
RADIUS
PREFERRED)
CUT
CUTTER
ROTATION
FOR "CLIMB"
MILLING
(BETTER
SURFACE
FINISH,
GREATER
ACCORA¢-Y)
_
Figure
Machimng
7.2-1.
I$ogrid
With
Unflanged
Hibs
7.0.002
1¸
CR16_
CUTTER
Figure
7.2-2.
Machining
Isogrid
With
Flanged
contribute
to
structural
therefore
be
about
0.75
cutter
diameters
The
smaller
rotational
resistance.
the
Cutter
pocket
_lepth
or
require
a light
is
same
as
its
movement
diameter
0.5
as
finishing
shoald
an
absolute
pass
at
minimllm.
high
cutter
speed.
When
the
pocket
path
stops
on
triangle.
corner
a point
During
wall
of
the
pocket,
cuts
one
of
the
stiffening
by
making
the
corner
cutter
centerline
rates
especially
radius
before
this
the
feed
Ribs
fatigue
times
OVERHANG
resuming
"dwell"
radius
corner
also
in
numerical
This
cutter
the
holc
undesirable
larger
than
a small
contribute
controlled
the
especially
enlarging
ribs.
describes
the
a cutter,
chatters,
then
at
the
the
machining.
7.0.003
cutter
another
leg
a small
one,
drags
it occupies,
better
and
is
radius.
at
the
down
situation
radius
to
radius,
best
The
each
,Grner.
finish
and
cutter
of
the
against
under-
avoided
path
of
Slow
cutter
accuracy
the
tilt"
llan._od
ribs.
lo oo
Figure 72.3. A T_nk Wall ConhOt,,tt_on
7.0.004
weight
order.
Electrical
provide
a means
of
electrode
determines
may
are
an
cutter,
is
in
an
almost
cau*.ioned,
raising
or
achieving
s,ach
refinennent.
Beca,ase
the
shape
of
an
impressio_l
:ather
range
of
shapes
unlimited
however,
configurations
nor
is
complete
the
rib
intersections.
Figure
discharge
to
apply
as
keyways
elimination
of
Some
this
the
electro-chemic_l
can
versatility
produced.
with
restraint.
spliced
holes
hole
with
attendant
are
the
than
be
and
practical
machining
of
a rotating
The
designer
Sach
not
stress-
recommended;
sharp
recommendations
shape
corners
at
shown
in
are
7.2-4.
CR169
_.--SHAPE
OF ELECTRODE
\
_
/
\
MAINTAIN
__
[
I
]I(
PROr_ORTIONS
_"_'_"
OFCORNER
RADIiAND
._"
I I I
J•|
HOLE
IN NODE
K
I ! I._
I
"AS
MACHINED"
"_
//_:;
SHAPE
Figure 7.2.4. Eleczrical Discharge (EDM) or Electrochemical (ECM) Machining
to Reduce Isogrid Nodal Size and Weight
7.
3
7.
3. I
One
FORMING
of
Power
the
Brake
more
con:mon
progressive
pinching,
accomplishes
its
curved
surface,
Forming
provesses
for
in
a power
objective
by
suct'essively
_-1.
The
Figure
7.
brake.
formin_
Still
line
7.0.006
sin_ply
curved
el ft,¢'tive
(reasing
elerm.nts
and
line
may
often
elenlent
I)e
parall..I
surfaces
used.
is
It
s c_l" tht.
as
in
7.0.00(_
the
case
of
a tapered
The
of
a cylin<lcr
or
in
this
type
fracture
of
the
material
without
properties
apply
since
material
springs
back,
forming
is
for
stiffening
each
if
radius,
internal
tops
support
the
necessary.
expected
the
the
plate
likely
to
of
ribs
the
but
The
highest
ratio
This
was
the
tract
NAS
8-26016.
or
the
the
of
ratio
surface
and
adeq:lately,
waffle
shell
is
can
internal
7.3-4
shell
Fortunately
aircraft
unflanged
incorporating
from
and
7.3-5.
tends
rib
of
9.
the
far
and
the
While
size
and
should
be
noted
o¢
the
Thor-Delta
060
to
9.080
in
this
they
column
_-ase,
avoid
the
accompsome
straightened
of
that
the
of
deformation.
required
depth
the
metho:t
capability.
compression
plate
percent.
geometry
amount
vehicle
to
were
buckling
on
Con-
was
3-5.
the
1.5
NASA
forming
The
upon
pressure
these
is
7.3-2.
depend
external
ander
Figure
It
formed
alleviating
tested
be
Removable
process,
7, 3-3.
final
high
in the
maintain
The
widening
backling,
was
1.04
pertent
curvature.
isogrid
to
within
designs
pockets,
the
to
optimized
stay
attempted
in
may
problems.
so
external
over-
ratios,
in
the
difficulties
higher
su,ccessful
were
ribs
"bridged"
ribs
after
developed
of
these
temperature
carefully
Local
cylinder
Figure
deformed
at
of
been
Figare
in
radius
either
deformed
configtlration
stiffening
Figures
ribs
1. 5 percent
inlersections.
have
blocks
shown
be
ribs
nodal
depth
The
easily
assumed
forming
buckling
a compression
quite
ribs
to
radius
slightly
e.g.
aboat
to
were
straighten
the
general,
plate
ribs
to
In
process.
of
that
be
the
support
is
Room
therefore
complicate
no
formed.
permissible
m_tst
process
accentuate
on
being
shape
the
The
pockets
they
is
unpredictable)
at
can
forming
somewhat
s-lbject
with
the
a cone
(and
are
lished
of
on
considerable
exceeds
crack
in
conditions,
the
are
of
formed
depth
ribs
ased
final
confignration.
blocks
the
they
limitation
elongation
are
as
wing.
fundamental
ribs
converging
tend
in
these
to
be
cases
typical
geometric
deeper,
flanges.
7.0.007
for
space
boosters
limitations.
the
depth
_an
be
and
Where
simple
reduced
by
ot
7.0o006
CR169
PANEL
13 PLACES)
ISOGRID
WE LD
92.66
ID
" 188.5
4
Figure
7.3-3
T_st
IN.
Cylinder
CR 1C':
A
SECTION
A-A
I:::::::
0.745
0.755
0.428
0.448
UIA
,0.063
0.083
R
NODE
Fiyure
7.3.4.
Thor
Delta
I$ogrid
Geometry
1.0.009
TYP
•
7.0.010
.j.
Altho'agh
a
the
science,
brake
tile
quite
good.
skins
for
This
the
in
Figure
7.3-
An
additional
this
method
the
In
Age
For
example
of
to
performed
the
the
Saturn
have
successful
of
been
an
S-IV
and
quite
large
is
the
40-foot
power
may
be
applicable
be
filled
art
made,
production
work
this
that
should
method,
and
of
S-IVB
and
and
powerful
brake
than
are
waffle
the
crew
equip°
illustrated
creep
shape
at
noted
that
the
which
forming.
the
this
by
process,
is
roll
with
a
suitable
treats
the
material
more
gently,
and
may
produce
parts
with
the
part,
usually
alumin-_m,
consistency,
is
the
When
heat
treatment
simultaneously
and
the
a
standard
not
individual
part
to
elevated
temperature
until
Allowance
for
some
springback
the
part
formed
and
more
parts
by
such
respects
forming.
For
filler
material
Forming
is
developed
fixture
some
members.
accuracy
final
be
the
forming
curvature,
this
MOL.
pockets
and
more
be
verified
isogrid
of
promises
must
been
method
buckling
advanced
should
development
of
in
1.
A more
are
a few
boosters,
An
Creep
creeps
after
forming
prevent
pound
has
shell
needed.
7.3.2
is
Thor-Delta
is
to
process
res.alts
compartment
merit
forming
be
it
aging
part
said
to
two
be
age
'day-to-day'
processes
formed.
process
It
but
one
that
user.
formed
is
creeps
to
is
made
on
is
temperature,
com-
it
is
clamped
its
to
a fixture
predetermined
in
the
shown
in
and
final
shape
of
Figures
the
held
shape.
/
fixture.
7.3-6
and
/
at
S,l,'h
/
/
a
3-10.
/
/
In
this
case,
a
segment
of
a cylindrical
wall,
it
was
found
that
by
applying
springback
/
/
was
practically
force
along
small,
eliminated
the
edges
allowance
of
would
accurate
production
different
than
its
process
may
require
(preferably
in
steps
over-aging,
ing
offers,
and
or
among
the
part.
have
to
part.
the
be
made
the
same
point
staging,
rough
other
advantages,
the
machined
shape
for
must
assume
a final
shape
a
either
'ixture),
initial
final
(e.g.,
forming
an
tensile
in
sevezcl
appropriate
,_ith
the
opportunity
7.0.011
elongation
120-degree
with
stresses.
a tensile
the
in
part
starting
multiple
improved
Though
Where
machined
by
_:onsistency
arc},
creep
to
an
q'aite
the
forming
constd-ratio;_
power
was,/
brake.
Cr_-p
reduce
residual
of
form-
/
.iR1t;9
FiqL,r,'
/ 3(_
l,,()qr,d
in{_,,','i
_ F()rT_111)_l FIxllar,'
7.0.012
2zig-T37
Alumin,.lm
7.3.3
isogrid
Compound
Co,npound
because
means
that
is
a flat
parts
sometimes
successfully
formed
by
this
process.
achieve
cannot
machined
in the
with
be
flat
developed
are
distortions.
subject
Each
Each
curvature
form;_ng
offers
similar
in
integrally
stiffened
parts
shapes.
This
for
such
to
unpredictable,
configaratiota
has
special
and
m,lst
therefore
problems
be
requiring
solutions.
Machining
after
leaves
resid,lal
stresses
these
stresses,
causes
sible
to
hold.
fixttlres
as
Costs
well
ma,:l_ines
as
large
the
shown
in
large
missiles
very
small
d_l_e
an
accurate
achievable
with
actually
Figure
7.3-7.
bulge
moderate
are
rigid
difficult,
- and
probably
costs
of
relieving
if not
naultiple-
the
more
be
approximated
is
with
_runcated
cones
as
of
ogive
noses
o:a
of
aft
representative
geoctaetry
defhaes
taper
the
straight
conical
line
element
is
faired
final
conto'ar.
This
much
do-lble
curvature
an
accurate
co-sieur
isogrid
forming.
Whether
a question
forming
ha\'e
ts
flat
or
has
a single
curvature.
b,"
difficult
b, lt
the
problems
strtl,
tures
an:t
art,
i_'.
7.4
NON-I_ESFRt;CTIVI£
MANU
I.'A(:TI,
RING
io:_-destru,'tive
iso_rid
be
raised
Machining.
been
used
art,
peculiar
to
these
have
successfully
l'ortllin:-
tho,;e
_ss_iate<l
el
beexa
for_
coll11):)',111"]
with
end.
eno,_gh
to
A
pro-
should
be
is
circumstances.
power-brake
The
applied
to
the
forming,
and
strutture
which
curvature
ca'n
all
integrally
stiffela..-d
be
s-litable
Il_r
at
was
tl_ed
to
_t) ts,)grid.
INSPEt:
I'ION
A(:CI'|YI'ANCE
in_l),_'ctio:_
parts,
in
techniques
structures.
age
s.ach
to
manufacturing
and
ho!ding
elaborate
the
of
impos-
work.
s_milar
is
forming
lo,'ally
outside
standard
nlanafa_'tured
by
diagram
creep
S*.aT_dard
Tolerances
can
aircraft;
making
summary,
the
This
creep
worth
fabrication
doing
Even
_. lnachining,
amor_izatioz_
changes
or
Sabseq-ten
escalated
higher
of
contour
parts.
difficulties.
distortio_as.
are
capable
Fairly
ing
to
pattern
individually.
special
In
difficult
unrepoatable,
developed
been
Curvature
curvature
mainly
has
t"Ol_
tech_aiq,lt,
l,'luorcs,
7.0.013
s sho,lld
e_at
dye
pot_t, tra_t
_ept-
CR 169
SPUN
MONOCOOUE
NOSE
_. SYMMETRY
jL__
CYLINDRICAL
CONSTANT
CONTOU
R APPROX
BY
TRUNCATED
THIS
LENGTH.
FORMED
IMATE
CON_.S
PARTS
WITH
CURVATURE
OVER
INITIALLY
SINGLE
ON
FINAL
CONTOUR,
SLIGHT
BULGE.
POWER
BRAKE
INCLUDING
,'
ATTAINED
/
FLAT
TYPICAL
OF
MATERIAL
STOCK
SIZE
#
/
/
CONICAL
SEGMENT
PATTERN
.
/
PATTERN
BASIC
)
/
BYCREEP
FORM,NG
GRID
SECTION
O
s
"--.-...-_/,/_
ANDSIZEOF
/
POCKET_
D,STORTED
BY
TAPER
_
/
"-..'--CJ7
/
"
Figure 7.3-7. Contour Changes Using Cones
inspect
isogrid
penetrant
by
cracks
Delta.
in
which
the
Visual
5x-!0x
the
for
facility
automatically.
with
par_s
in
the
user,
S1VB
waffle
paris
inspectio;a
magnification.
individual
The
Other
e.g.,
dye
par_s
were
o.r the
were
han.'tled
handled
and
par_s
inspectio._
penetrant
skins.
"7.0.014
was
then
in
the
was
applied
sometimes
may
,ased
at_tomatic
penetrant
used;
techniques
which
an
be
developed
::o loo'<
for
forming
REFERENCES
I-I
I-2
Meyer,
Evaluation
R.
Douglas
Report
Knighton,
Analysis",
Presented
R.
and
R.
of Common
SM-47742,
D. J.,
"Delta
NASTRAN:
at the
Second
Center.
llampton,
2-I
Meyer,
Combined
Loading."
2-2
Timoshenko,
R.
Timoshenko,
Shells."
and
J.
N.
11-12,
Isogrid
Structure
NASTRAN
NASA
TMX-2637.
at Langley
Research
1972.
Stiffened
Normal
June
Goodier,
Sept.
Cylindrical
Pressure,
1972.
UCLA.
"Theory
of
1972.
pp
121-143.
Shells
Bending
Subjected
to
and
Shear
Elasticity."
2nd
of
and
2nd
S.
Ed.
and
Woinowsky-Krieger,
McGraw-Hill,
1959.
Tirnoshenko,
S. and
2nd
Ed.
McGraw-Hill,
2-5
Flugge,
W.,
3rd
Printing
2-6
_Aeingarten,
V.,
Walled
Circular
J.
"Stresses
lq66.
M.
Gere,
1971.
"Theory
in Shells,"
_.,
Seide,
( /linders."
"Theory
of
Elastic
Plates
Stability."
Springler-Verlag,
New
and
,l. P.
Peterson,
NASASP-8007.
Sept.
York
"Buckling
1905,
2-7
Weir
garten,
V.
and
P.
Cones."
NASA
SP-8019,
Seide,
Sept.
"Buckling
1968.
of
Thin
Wailed
Trtmcated
2-8
Weingarten,
V.
Curved
Shells."
and
P.
NASA
Scide,
SP-8032,
"Buckling
August
of Thin
1969.
Walled
Doubly
2-9
Ge'ard,
Part
ti.
2-10
Jenkins,
Isogrid
Number
G. and
Buckling
W.
C.,
of
Becker,
Flat
"Itandbook
of Structural
Plates,"
NACA
TN-3781,
"Determination
Stiffened
Cylinders,
MDC
G2792,
Feb.
of
Critical
" Mcl)onnell
1972.
R-1
Stability,
July
1957.
Buckling
Douglas
I.oads
Report
h_c.
of Thinre_,ised
1968.
I -
Ed.
1951.
2-4
August
Sept.
and
Experimental
Stiffening",
1964.
Launch
Y ehicle
User
Experiences.
Colloquium
held
Va.,
S.
"Fabrication
Waffle-like
November
R.,
"Buckling
of
Axial
Compression,
Ph.D.
Dissertation.
McGraw-Hill,
2-3
J.
Bellinfante,
Domes
having
f,_r
t
2-11
Petersoa,
Compression
Decembec
J.
P. "Buckling
of Stiffened
Cylinders
& Bending
- A Review
of Test
Data,
1969.
2-12
Meyer,
Buckling
Report
R. R. and H. A. Anderson,
"Space
Test.
Vol.
I, Design
and Analysis,
MDC G24804,
February
1972.
2-13
Gerard,
Part
III
G. and H. Beckcr,
- Buckling
of Curved
August
19 57.
2-14
Peterson,
Circular
TN4403,
J. P. and R. G.
Cylinders
Subjected
September
1958.
5-1
Rubinstein,
Prentice
Hall,
M.
F.,
Inc.,
in Axial
" NASA
TN-5561,
Shuttle
Isogrid
"McDoanell
Taak
Douglas
"Handbook
of Structural
Stability,
Plates
and Shells,
" NACA
TN-3783,
Updograff,
"Tests
of Ring Stiffened
to Transverse
Shear
Load,
" NACA
"Matrix
Computer
Englewood
Cliffs,
of Matrix
New York,
of Structures,
1966.
Przemieniecki,
McGraw-Hill
J.
Book
5-3
MacNeal,
R.
NASA SP-221,
H.
5-4
Heywood,
R. B.,
"Designing
Hall,
Ltd.,
London,
England_
5-5
.Jenkins,
the S'.ress
W. C.,
"Three-Dimensional
Distribution
m isogrid
Douglas
Report
MDC-G2496,
Jenkins,
Isogrid
March
W. C.,
Stiffened
1972.
"Determination
of Critical
Buckling
Loads
for
Eylinders,
" McDonnell
Douglas
Report
MDC G2792,
5-6
S.,
"Theory
Company,
Analysis
N. J.,
5-2
{Editor),
Septembet
I
' The
1970.
Structural
Analysis,"
N. Y. 1968.
NASTRAN
Theoretical
by
Pho*.oelasticity",
1952.
Photoelastic
Stiffened
Panels."
March
Manual,"
Chapman
and
Analysis
of
McDonnell
1972.
R.2
j
ADDENDUM
L
Section
LXPLklMENTAL
Analyses
of
_soqrid as presented
tile prediction
isogrid
of bucklinq
RESULTS
instability
and of states of stress
under the action of loads, singularly
of such analyses
structures.
Tt|ese tests may be either
tests J. subscale ,_dels.
and wIG_ data scatter.
_odeis,
Full-scale
that occur
in
metal
tests of structural
in combination.
on isogrid
structures
or
components
have
for theory verification,
slnall buckling
During recent years,
with
by tests conducted
them undesirable
t_L_deiro,_ Lexan polycarbonate
Tt_ese ddvanta_es
a.
metal
concerned
or applied
tests of full-scale
upon buckling,
,:_o,lels
_a_ sevL;ral auvantaqes
applied
can only be verified
ci_aracteristics which make
e._., pl,Jstic deformation
TESTS
in this handbook are primarily
The validity
certain
FROM MOUEL
"knock-down"
factors,
the fabric;ition and testing of
plastic,
have shcwn
over the use of full-scale
that the use of such
metal
test models.
are SUln,_arizedas follows:
Lexan ,_)dels are relatively
inexpensive
compared
to full-scale
_net_l models
I_. ir_e larqe ratio of strain-to-yield
el,_ti_: bucklinq
of the models,
stress of Lexan results
thereby
allowing
in
the repeated
testi_a of a sinqle model
c.
Past {)roqra_s with Lexan models
have bucklinq
nave shown
that such models
"knock-down"
factors
approaching
typlc,_l rletal "knock-down"
factors
of c).u to 0.7
,,. Past proqrams
have shown that buckling
r_aw neqliqible
data scatter
unity
versus
data from Lexan models
,'.
L_'xd,,,a, I_e ilonded witn so]vent adhesives (etny'lene dichloride
_(,thyI,,r,:
ch]orid(') thereby
f.
tl_e simulation
Lexan, unlike other
plastics,
not require
handling during machining
;,_is sectlon
special
is an extremely
tough material
or special
as presented
in Section
TESTS OF SKINNED
S_6JLCTLu
LOADING
Tu COMBINED
of the results
of the results
L_X,, '40JLL hbCKLi_G
3./. 1
of weld joints.
to a documentation
tests and a comparison
from analyses
of models
and does
handling
of
moaei.
wili be devoted
plastic model
_.2
the construction
,Is wull as a11owln9
tr}e comp|eted
results
facilitating
or
of these
of such subscale
tests with the
4 of this handbook.
AND UNSKINNED
OF COMPRESSION,
ISOGRID CYLINDERS
BENDING,
AND TORSION
{)ackqrG_nd
Generai_v,
a venicie
structure
will
_n the _,,Ifiliment of its mission
,,tabliltv oF a structure
be subjected
objectives.
under sucn
to a combination
As is well
loads is affected
t_e ind_v_d_<_l types of loads applied
known, the buckling
by the proportions
to the structure
si_ear, torsion,
in _nic,. t_Le ,oads interact
to affect
the buckling
stability
in :ntt.rac_o,: e.iuat_ons as presented
in Reference
8-I.
t:_e _nteraction
equations
i_,_porL_:;'.
:o ._';,_ssthe accuracy
r_,,.,:_
>,:, _ ;_qrld
and internal
are empirically
of such equations
cylinders.
2
of
at any given moment.
i.e., c÷_;press_on, Oending,
_ndic,tes,
of loading
pressure.
The manner
are characterized
As this reference
determined.
It is tnerefore
in tne prediction
of ti_e
,,. ,_ _,,, l,.,i
_,.rLI i(.,_ti,_ L_I ',_t:_l interaction
ct_|sistlrl(| ()l tile design, fabrication,
:_,i_,
wi_;_out skin
of bucKlinq
(unskinned)
equations, a prograln was conducted
and testing of two isogrid cylinders.
and one with
skin (skinned)
i,ehavior under the action of compression,
for the determination
compression-bending
and
torsion.
,,._.2
Des_qr, and Fabrication
of Models
L_oti_cyllr;Jers were desig_e(l to buckle in general
accoL_;alis,_edusing the analyses
sKiu_ed
cylinder
given
for compression,
and for t,_e unsKinned
cylinder
in Section
instability.
4 of this handbook
compression-bending,
for compression
isogrid cylinder
sis ,.it,
velope_i ',_y;4cDonnell Douglas
Astronautics
In all cases,
ta_.et_a_ u_t,l.
isogrid
ri_ stresses.
fiquration
Circumferential
t_:+tor._ary
units were used
I_o attempt
,.+>c.,_.
-" LJnits _d
was determined
Company.
cylinder,
...,.;,v ' .,,,"
,' ,r_,_ ;_to curved
i.Dr a qiven
,. :r;,,,,_'_',, _(re
LUG;,.*
con-
either cylinder.
all results
in both the International
of the machinir_g of t_e appropriate
,, ,r.,,,,;(::_
:,+_'.it_,)_
_ t_r,_e flat plates of Lexan.
.+
to minimize
isogrid
In this section,
are given
employed
i_ custol_ary units.
,,,....,-_,:._+i..,
oT eac_ ,:ylinder consisted
.,.;,, ,::.
by an analy-
factors were
an external
was made to optimize
calculations
buckling.
All equations
the "knock-down"
in all calculations.
._f ,_.xper_:,(:nL,+._
and theoretical
buckling
was used in both cylinders
For the case of the skinned
was employed.
and torsion
for the
and compression-bending
;ors_o__,i)uc_lin!lof the unskinned
,:_-t,
,,_+_:_:_,_._ze,i
in Table ,5.2-I.
This was
These
plates were
configu ration_, constituting
cylinder,
solvent bonded
three of these curved
pieces,
SUCh that each cylinder
;,;_'L:,. T_,) pr,ilosopny of the desiqn
one-third
of a
hereinafter
had three
of the longitudinal
sub-
Joints
Iongiwas
IIE IIODUCIBII, OP THE
ii, IOI Al., PAGg m POOR
t_ dt,,._,:r.
,_ )(_int Lhat would
,,,,_i,.
....
L,J]_coL()type.
s,(_v_ Jr_Fiqures
realistically
The final measured
_).2-I and 8.2-2.
for tr,:se dimensions
simulate
dimensions
The calculated
using the equations
in qeneral
J.2. J
Test Set-up
instability
Gotn v_odels were mounted
cylinders
a distance
before
of the cylinders
buckling
stress
were conservatively
end fixtures
of 5.08 cm (2 inches).
To assure
fixt,Jres.
Compression
and compression-bending
cylinders
by a compression
in conical
depressions
test machine
were applied
acting
through
end fixture.
a steel
Application
of the end fixture
resulted
the cylinder.
compression-bending
of the cylinders
quarter
the radius
at depressions
(R/4) and one-half
_or a r_oz_ocoquecylinder,
such off-axial
ratio ot u.b af_u 1.0, respectively,
f
C
i_ tr_e axial stress.
the radius
Torsion
t_)rOuQh J_ionlerlt
arms on the upper end fixture.
were applied
is shown in Fiqure 8.2-3.
lOd,J cells were employed
in monitoring
sphere located
of the load in a
was achieved
(R/2) from the cylinder
axis.
correspond
The manner
to an fb/fc
bending
by a hydraulic
the loads applied
f,_r t,,e axial load and two for the torsion
loads.
t:_eodtputs
recorded
stress anu
jack acting
in which
these
loads
three
to the cylinders,
During cylinder
one
loading,
on a dual pen X-Y recorder.
REPRODUCIBILITY
ORIGINAL
4
by
one-
As also silown in this figure,
of these cells were continually
to
located approximately
fb is the maximum
load was applied
to the end
in pure compression
loading would
where
of torsion
to the test
,lepression at the center
loading of the cylinders
to
into the
the transfer
end bands were used to clamp the cylinders
off-axial
designed
that penetrated
loa_s to the cylinder,
Combined
in Table _.2-2.
eitner rib or skin buckling.
in aluminum
in tile upper
are
resultants
of Table 8.2-I are shown
,ks can _,eseen fron this table, Doth cylinders
buckle
the weld joint of a full-
PAGE
OF
TIIE
IS POOR
. .,i _;_.:_';;.;_.rlce,
r<,:_al_.s,
,,nd Discussion
k,,T.,_
(yli_ders
were
ir_itially buckled
,'orslon to deter_.ine bucklinq
_ucklin,l patterns
in pure axial
behavior
for tt_e skinned
of Results
for these
cylinder
compression
loadings.
are shown
the post-buckle
cylir_der _jT_derpure axial compression
indicated
Subsequent
dual Dea_: osc_,ioscope
and electrical
tnat tne buckles
initiating
were
inoicazes
t_at considerable
t,_rsion._ _ucK_ing'.
A subsequent
deformation.
re_.1_r,),i
by solvent
bondinq,
average _ximum
7bb7 h (1699 pounds)
compared
to a maximum
To prevent
further
torsion
load could be immediately
preventing
the cylinder
the n_a_i_:,_,_
torsion buckling
indicated
concern.
Figure
The
8.2-5
in
of this cylinder
indicatea
that
axial
existed
the fractured
load after
that other
rib was
rib fracture
load of 8095 N (1820 pounds)
to this cylinder,
released
a
occurred
was configured
from undergoing
in the
cylinder
Although
damage
_sed to apply torque to tne cylinder
centers,
and the possibility
plastic
of the skinned
rather than in the joint.
to the skinned
r_bs had experiencea
rid fracture.
at buckle
of no further
examination
one circun_ferential rib had fractured
pattern
and
tests of the cylinder, using
contacts
deformation
8.2-4
that buckles occurred
in the isogrid
_uc, les i(',
the joints were therefore
The resulting
in Figures
._..'-b. As can be seen in Fiaure 8.2-4
jnints as well as in tne isogrid.
and in pure
at the onset
before
the hydraulic
in such a manner
system
that the
of buckling,
large displacements.
was
thus
Interestingly,
load was found not to have been affected
by the
rib fracture.
,x}al _ucKlin!l of the unsKinf:ed cylinder
indicated
.,. ,.:_,_
.iolnts.
by the bonding
T_is problem
was overcome
,,_i_(,_ce _;.Z-C, to t_e _nside and outside
_e
effect of qreatly
increasing
_r_a_ uuckling
of the joints.
the bending
stiffness
was initiated
of longerons,
These
longerons
of the joint
without
REPRODUCI]3II,ITY
ORIGINAL
shown
PAG/_
nad
a
OF' TII_';
IS PLE)R
.,,r,.",_,,r_,l_f,, _1_(re,_',,,in extensional stiffness.
Measurementsmadeon tensile
sample_, desiqned
to simulate
that the addition
double
increased
lon_erons
the joint,
indicated
the effective
by approximately
53 percent.
isoqrid cylinder
for pure axial
extensional
The buckling
patterns
No damage occurred
bucKllncl as nad occurred
for the skinned cylinder.
quent tests for torsior_ buckling
system previously
A total of 54 bucklinq
E_ucklinq interaction
tests were
of combined
w_1_le a×i._i load was applied
performed
from zero to 100 percent
percent
increments.
zero Dendinq,
Axial
of maximum
torsion
loads were applied
and at two off-axial
coi:_pression-bending.
occurred.
To prevent
along
load points
anomalies
all subsethe quick
cylinder.
to determine
compession-bending,
and maintenance
of a torsion
Torsion
load
loads were
load in approximately
the cylinder
axis,
20
for
(R/4 and R/2), for combined
in the data from local trisector
w_re performed
and tne results
The data from these tests are summarized
tt_e(iraphs of Fiqures
r,ltio_
,).2-9 and 8.2-10
(_st = i_t_st
to_-sior;_,_cK_q
bucklin 9 stress)
as functions
buckling
and averaoe
wi_ere fst is the torsion
stress)
for buckling
in terms of average
,C _ f/F c wr_ere fC is the compression
_;_axi_U_:_
compression
bucklino
torsion
on each trisector
compression
in
stress
stress and F C is the
stress ratios
stress and Fst is tne maximum
of the compression-bending
6
the
and torsion.
variatior_, a_l load combinations
were averaged.
in
from torsion
employed
on each cylinder
compression,
until buckling
varied
cylinder
unskinned
are shown
Nevertheless,
for the skinned
TJ_is was accor_,_lisned by the application
loading
to this cylinder
of the unskinned
described
of the joints
of the modified
loading and pure torsion
Figures _].2-7 and 8.2-8.
load release
stiffness
of the
ratio fb/fc.
It_,l,,t_.rmii_r'
lh(' _,ff:'(
I (_t th(' order
in which
Inads werp applied,
both cylinders
_,,i,.._I..,,
l_,,L,h,,l
bV ,il,l,
li(:,Itiorl
of axial compression
foll(lwed by torsion.
:-,_,ultsof this reverse
identical
loading
load order were
essentially
The
to the original
order.
Reference
torsion
8-I gives an interaction
equation
for compression,
and
of
Rc + Rb + Rst
2
=
(8.2.1)
l
This
where Rb is the bending
Figure 8.2-II
stress
eYcellent
correlation
experimental
curves.
curves for fb/fc
values,
correlation
i.e., Rc
is plotted
A comparison
of Figure
and 8.2-I0
in
results
are defined
0 and Rc
results
because
to be the same as the
0 at Rst
I.
In like
the same for both
ratios of fb/fc.
curves may be equated
end points by modifying
that
the theoretical
0 is by definition
for the two remaining
indicate
ratio of zero but that the
for fb/fc = 0 partially
l at Rst
The e,d points of the theoretical
experimental
8.2-9
for an fb/fc
the end point for Rst = l and Rc
theory and experiment
equation
of 0.5 and l both lie above
the end points of the experimental
theoretical
of O, 0.5, and I.
data of Figures
is obtained
The excellent
interaction
ratio.
for ratios of fb/fc
8.2-II with the experimental
manner
bending,
Equation
to the remaining
(8.2.1) as follows
2 = l
k(Rc + Rb) + Rst
(8.2.2)
where k = (Rc)th/(Rc)ex p and (Rc)th and (Rc)ex p are the Rc values
for theory
and experiment,
ratio of
fb/fc •
r_su_ts
respectively,
A comparison
indicate
for Rst = 0 and for the appropriate
of the results
excellent
of Equation
correlation
as shown
(8.2.2) with the experimental
in Figures
8.2-12
and 8.2-13
for all ratios of fb/fc for both cylinders.
_CL," :'
....... ' ":I_IT
"'"' I'"_' C,iF,"Ftl t,I
i
;',_i', v\_ ,'_ I_'F_t ,t,lv't','l:,'tll.
L,lUatlon
(_l.?.l)
Lylind,:rs.
is
There
Jsl,li_,_Ll_:,
essentially
are
th(IL
Lh(: shape' of
correct
a number
These
of
for
reasons
skinned
and unskinned
wi_y the original
data
_ith
correlation
between
theoretical
and experimental
buckling
for
pure
,_xial
conlpression
and pure
torsion.
The practicality
of
the
data
that
only
the
to
completely
curves
be ascertained
need
in
order
later
from
deal
is
in
varies
tnat
adjusted
be mentioned
isogrid
values.
of
will
by
te_e theoretical
corrula_ion
reasons
both
Lh(' curvt,_, _iivt-'n
paragraphs
the
end points
of
define
entire
the
the
loads
excellent
interaction
interaction
curves.
Tlle averaqe
values of axial
_or the skinned
the torsion
load were
and unsKinned
755? N (1699 pounds)
cylinders,
respectively,
loads were 479 M-N (353 ft-lbs)
skinned and unsKinned
cylinders,
_alucs _vith the theoretical
of Table
total loads to equivalent
stress resultants.
cylindPr
a portion
joints
reduction
supported
is relatively
axia_ strain may be assumed
a uhiform
axial strain,
(Icr
_r_ere ,
=
The comparison
_.2-2
require
for the
of these measured
of the
in that the
to the cylinders.
for exial compression
by virtue of the rigid aluminum
the stress resultant
values of
the reduction
Tills is necessary
of the load applied
straight-forward
and the average
and 353 M-N (260 ft-lbs)
respectively.
values
and 7167 N (1611 pounds)
in the isogrid
in that a uniform
end fixtures.
may be shown
P
cr
_
t (l + _)
is t,_e critical
buckling
previously
force
per unit length,
area, Pcr is the total bucklinq
defined
Given
to be
(8.2.3)
A is the total effect-
r
_vt. exte,sionai
This
in tnis handbook.
8
load, and t and _ are as
APi_l},,itiL_l
_! tht. ai_l,r()l,ri,_t(_
values from Fiqur(_ ;'.;'-If(_r the skinned
and fr_m i i_iLir¢'
_i.;'-;'
f{)r the u_iskinned cylinder
N
(skinned)
Ncr (unskinned)
(0.417 M -l) Pcr
=
load of 7557 N (1699 pounds)
for the skinned
an ;_ of 550C ;(IM (31.4 Ib/in) corresponding
cr
values of 7230 _/H (_I.3 Ib/in).
carried
by the skinned
from torsion
load.
unskinned
cylinder
tile theoretical
A comparison
values
to 76.1 percent
As previously
This
iqad corresponds
an average
axial
mentioned,
then give
of the theoretical
the maximum
load
to 81.5 percent
damage
of the theoretical
load of 7167 N (1611 Ibs) for the
gave an )icr of 2990 N/M (17.1 Ib/in) for 89.5 percent
of
values of 3340 N/M (19.1 Ib/in).
of the theoretical
torsion buckling
is not as straight-forward
relationship
cylinder will
cylinder was 8095 N (1820 Ibs) prior to cylinder
loading.
In like manner
will qive
(0.728 M"I) Pcr
cr
An average
cylinder
between
torsional
the joints
is _ot known.
in torsion
at a critical
as the axial
strain
Ocr(1) where ,)or(1) is the critical
ship can be aerived
case just treated
shear strain and the torsional
If the assumption
shear
loads to the experimental
that relates
is made
(i.e., Vcr(1)
shear
strain)
aefined
as the shear strain
torsional
rigidity
ot d 360-degree
that the isogrid
determined
by
buckles
is the proportional
per unit torque,
isogrid
load carried
then the following
the experimentally
rigiuity,
in that the
to
relation-
torsional
to the theoretical
cylinder.
(8.2.4)
_ne-_ T _s the expected
rlgi_ity,
torsion
load,
(OIT)th
(_,/T)exp is the experimentally
is the theoretical
determined
9
torsional
torslonal
rlqldlty,
a,ld
!
I,,,,
_,,,,(_r,,Li(.al
If,._)r_'ti_:,_l
torsional
Lrll.
torsion
1.19 Tcr
T (unskinned)
=
I. 36 Tcr
tne calculated
Ri::PZ-_oDUCIBILITY
OF THE
ORIGL-NAL PAGE IS POOR
value of Vcr is 1460 N/M (8.31 Ib/in)
to a Tcr of 374 M-N
(276 ft-lbs),
load of 445 M-N (328 ft-lbs).
a value
1_ke mariner, tile calculated
and
will give
=
4GO _.I-_(354 ft-lbs),
for a T
APl)licatiorlof the measured
load.
T (skinned)
corresponding
buckling
torsion
rigidities
For the skinned cylinder
(Table _._-_)
l(,lI
8 percent
The experimental
greater
giving
value
was
than the expected value.
Vcr for the unskinned
of ?04 M-N (151 ft-lbs),
for an expected
cylinder
an expected
In
is 786 M/N (4.49 lb/in)
torsion
load of 278 M-N
cr
<2t)b ft-l_s).
Actual
titan tne expected
torsion
load was 353 14-N (260 ft-lbs),
value.
There are a number of buc_ling
between
considerations
tne ti_eoretical and experimental
ana tne discrepancies
interaction
27 percent greater
between
that relate to the discrepancies
compression
the theoretical
and torsion
and unadjusted
loads
experimental
curves.
The b_ckilr_q equations,
given in Table
hanJL_c,,, ,_s_u,:,e
simply
supported
.v) f_c)t, _s:.:,_,"one effects
,._(:._;c_,signs Jf joints,
8.2-I and taken from Section
end conditions.
of prebucklin9
or the festoon
Furthermore,
bending,
curve
predicted
,r,,)r_,erto transn.it tne torsion
to the cyllnders.
increase
:_e buckling
loads for both torsion
lO
rigidity,
by tl_e Fl{Jgge equations.
the ends of tne cylinders
loads
4 of this
these equations
rib torsional
i_)r t u ;,,
aeis tested in this program
will
buckling
were clamped
Clamped
and compressioh
end conditions
over
simply
REPRO!'_
'.==i'P,-=,.,l
,.,_,i ,,mdltl(m'..
l_t" Llle
F,".p_Jli',ll,i,'
8
percent
_.2-5,
qreater
tim torsion
For the unskinned
did not occur,
cylipJer
It
i',
_?xill,rilli_'flLdl
buckle
that
torsion
load of
pattern
cylinder,
thereby
Ti_e festoon
curve effect,
in the skinned cylinder.
mentioned
all buckles
For example,
buckled
4.2, as well
buckling
to torsion
responsible
for pure axial
for the displacement
data away from
the theoretical
cylinder
off-axial
under
tnat this ratio of compression
riqidity
of the isogrid
capa:_ility of an isoqrid cylinder
equations
of this l_andbook.
increase
in
com-
loading and have little or
interaction
loading
bending
:,acklin,.l
_,_ad of t,_.7 percent of tnat for pure axial
;,_e torsiorml
in conjunction
as prebuckling
ribs contribute
but t_is effect
In the majority
would nave a minor effect as the cross
curves.
but classical
should result
in a
compression.
to the load carrying
is not included
in the
of cases, suc, torsional
sectior_ of typical
II
of the
for fb/fc = U.5
at °9.5 percent of tile load for pure axial compression
t,_eorb'iudicates
of the
(compression-bending).
that these effects were
tl_e skinned
in the joints
value.
the compression
for off-axial
of the
portion
in the 27 percent
in Section
no effect
experimental
to the isogrid
that this restriction,
resulted
These effects do not apply
unadjusted
stiffness
to the point that buckling
pression.
It is thought
hand, the bending
load over the expected
loading
being
a joint
ends,
will tend to decrease
the skimled cylinder
were
traverses
It is thought
bucklinq
end conditions
As can be seen in Figure
restricting
tne torsion
clamped
load.
on the other
the effect of the clamped
these
torsion
by the longerons
(Figure 8.2-8).
bending,
Li_ouqllt
tt_a,_ti,e expected
joints was increased
with
:_.'LBII._'I"v 0!:' '['H e;
isogrid ribs
rigidity
is
I
_-,_'-'_I',
_"
"_,-_II
i_'_
•
• ;o
°
rei,iIiv,,Iv _,mali.
For the unskinned
v;iutt_of the ribs was approximately
__)ivln!_
tilu ribs of tz}is cylinder
ti_at _t_t_iflcrease in buckling
r_,iaLlveiy _i1gr__i
Cr
cylinder
of tills program,
significant
tbtsional
load by tills torsional
cylinder
rigidity.
rigidity
paragraphs
in reia_on
to tne effects
of clamped
provided
ends and rib torsion
that buckling
in relation
_+;,;ct ,IT:,I
t:.L _ffects
bending.
of prebuckling
Inciusive
In practice,
presented
t,_ t.,is ",",L >r,.,.';raF._,
as well as the specifics
,,.:;._n CSe_tion 8.2), are presented
;
r'.,:nce
_
in the
festoon-curva
account
of structural
is
i,,_per-
i)y tests on
in this handbook
as related
of the test program
described
in detail
in the engineering
test
_'
• tn_-,<-xp,:r_;'_(:l;tal
program
,.,. ....
and also
,_etal _tructures.
_._,,_:e,v,,
_,'i;,_rtures
from the theories
"
(prior to
rigidity
to the FlUgge
factors determined
"kr_ock-dowll"
re:_,'..,_,'nt,_t}v?
full-scale
,, r.
value
is not initiated
",,,_,.-,_':_.
t,_,,
of tr.,,,__:ff(,cts,as well as the effects
;r_.
t.i,
of the calcu-
cylinder.
_.c,-nt._._ c t._eory is unconservative
'L_' ai]
in the
indicate that the theory of Section 4 is conservative
.;or tr'_.effest_ of joints,
l__t_,_:_';
It is thounht
resulted
of 89.5 percent
l,_te,Iv_iuc versus the Ncr of 81.5 percent of the calculated
T._,:prcvious
the
90 percent of the depth of the rib, tilereby
for the unskinned
r'ib Ja_,_a!;e)
or the skinned
however,
,,_..,_,_,.',._'._)
adequately
,.....,.,',,,u,r,,; (.;v,;r.._ers
subjected
indicate
describe
_l_at tne interaction
the response of skinned
to combined
compression,
,,_,i;.I_r_:,
,iil,i
L,_rSlOF.,_rov_Jed the end points of the theoretical
12
curves
and
compressioncurves are
l
Irl practice,
_',it_,_L,',l
I,J t,,,' ('xl),'r'li,_'i,L,Jl
ddLd.
a
proposed
i,,,r'I_n,a,l _)11i;i'.
td_
"_;:'.;t
il.lt_l_S
Of
.JnJer pure torsion
and under various
Ja:,,_/roJL_these tests combined
£t
sr_o.iJ
also be reallzed
that effects
inf1_(_nce t_e bucklinq
stability
tffects are qenerally
handled
rei)rcsL'ntat,ve full-scale
combinations
define
other
loads
The
described
curves.
than those analyzed
of a configuration
13
tests must be
in the manner
the interaction
structures.
that
of compression-bending.
(8.2.2)
by a "knock-down"
metal
llleans
copfigur'ation for buckling
with Equation
_n :,cction 3.2.4 _vill completely
this
and that these
factor determined
in Section
secondary
by tests of
4
REFEREI_CES
'.;ru:,_, !. i.,
"Analysis and Desiqn of Flight Vehicle Structures,"
ir1-_I,_',_.
_ifset Company,
Cincimlati,
Ohio, 1965.
,_,.i.._. , ., "Lxper i_ental Oetermination of the Buckling Interaction
,,f .',Ki,,n,_.,1
and linskin_('dIsogrid Cylinders Subjected to Compression,
_,_,_p,._,,,_ion-_,endin(l,
and Torsion,"
.',uqu
st I,!74.
McDonnell
14
Douglas
Report MDC G5238,
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REPRODUCIBILITY OF THE
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