A previous analysis of fluid filled storage bags is extended to the case of a long fluid filled cylindrical membrane supported by uniform line loads. Cross-sectional shape, stiffness of the support system and stress resultants in the membrane are determined. The application of the numerical results to problems arising in the design of nonrigid airships is discussed

Duncan, J. L.

Mullins, M. L.

English

NASA Technical Reports Server

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N76-15033 j
LONG FLUID FILLED BAGS
SUSPEI|DEDBY LI[IEFORCES
' N.L. Mullins*
J. L. Duncan**
ABSTRACT: A previousanalysis of fluid filled storage bags is extended a
to the case of a long fluid filled cylindricalmembrane supportedby
urliformline loads. Cross-sectionalshape, stiffnessof the support
system and stress resultantsin the membrane are determined. The
applicationof the numerical results to problems _rlsing in the design
of non-rigidairships is discussed.
ItlTRODUCTION
Long fluid filled bags are used for a variety of purposes and examples v;hichhave
been studied include sausage-likestorage bags for oil (I) portable silos (2),
inflatablestructures includinglife-rafts (3), suspendedcylinders(4)anda variety
of non-rigia pressure airships,"blimps",and semi-rigiddirigibles(5).
The long filled cylinder restingon a horizontal flat base was considered by Demiray
and Levinson.(1) They obtained a solution for the stress resultantsand the shape
of the bag in repose. I_ this presentwork, their analysis is employed and extended
*GraduateStudent, Dept. of Ileci_anicalEngineering,rlcIlasterUniversity,Hamilton,
Ontario, Canada.
**Professor,Dept. of HechanicalEngineering,Mdlaster University.
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_ to apply to the case in which the long bag is supported by concentrated loads applied
- Lu the membrane along lines parallel to the bag axis. It is shown that the results
__ can be summariTedby functionalrelationshipsof non-dimensionalparameters and some
_{"_,.._ numerlcr,l results are presented.
_ : " The solutionsare relevant to the design of non-rigidpressure airships. In these
/J:_ vehicles the principalfixed weight, the car, is attached to the fabric envelope by
_, I a so-called"catenary"suspensionsystem inside the envelope as shown in Fig l
_ CATENARYSUSPENSION _"r_
¢
6
IL' "_
" CAR
,' Fig I General arrangementof a non rigid pressure airship
"_,._ The envelope is maintainedat _ constant differentialinflatlonpressure by pumping
" air into the ballonets shown. The fabric is reasonablylight and woven in such a
way as to resist both direct and shear strains. The inflationpressure is sufficient
to maintain the shape of the envelope under static and aerodynamicloads. The
applicationof the numerical resultsof the two dimensionalanalysis to the case of
airshipenvelopes Is discussed.
!.
! THE AI_ALYSIS
., The membrane is assumed to be Inextensiblein all directions,to have zero flexural
_. rigidityand to be weightless. We consider a normal section of an infinitelylong
uniform bag. Under equilibriumconditions,the cross section is representedby the
:. curve.
x • x (s) x(o)- o
y - y (s) y(o) • o (1)
where (x,y) is a set of rectangular cartesian coordinates and',; is the arc length
measured from the lowest point, the origin, in Fig 2.
: 2OO
1976007927-207
I ' '"' .?" II II II IIII _
The stress resultantin the membrane is T, which y
is constant between loads, and the angles between
the tangent to the membrane and the horizontal TI
is O(s). For the cases to be consideredhere, the /__
hydrostaticpressure or more strictly the differ- /
ential pressure across the membrane, is p(s) which P
is taken as,
p(s) = p o+ wy (2)
where Po is the inflationpressure at the lowesti_
-: point,y = o, and w is the difference in specific
:_ weight between the fluids insideand outside the
membrane. In general w may be either positive or
_" negative and in the problems considered,it is
: constant i.e. the fluids are incompressible. Fig 2 Membrane Coordinates
; The general solution to the problem for w < 0 is given by Demiray and Levinson(I)
_ Define R as 1/27 times the perimeterof the membrane cross-section. The following
dimensionlessgrouping:will be used:
_- X , T
_,_ For conveniencein writing the followingequations define
"' k' = 4T._.ww= 4 _ (3) ,
:; p,= (po/Rw)
' For (4Tw/p==)>O, Ref (I) obtains !
: _= _L / [l+k'sin'(B/Z)]- 1} (4_ !
x T sin e !
_ "R"" 2 _ ¢(i_k2sin=(OlZ)] _-
- 41+k')E ',,'(I+-T;F
T +_1 1 E_, j(l+_ ] (5)i
S=ZT I F k
.- II _ _ ' ¢(l+k=) (6)
where FI_,p] and E[_,o] are the e111ptlc integralsof the first and second kind :_
i
_' 201
i
1976007927-208
..................... I"/2 .
- . respectively, and = is deftned by ."
7
[Vll „° stn(./Z)l' = = arc sin (7)
._: Previous work did not consider the case for w < O. It may be shown(6) that for
-I < (4Tw/po2) < O,
s. _ _x=2_T [(I-_ ,k ,_Z E ] (a)
"_. R p°R k2 k2
" R pQR
$
i For (4Tw/po2) < -1, [,;]x T I., -= 2 [2E , - F , ] (10)'_, R poR k
l
.. s__=zT__1F , (Ii)
R poR k
. where ¢-arc sin [k sin(e/2)] (12)
> For both of these cases,
Y = P_._o{/ [l.k2stn2(e/2)] _ 1} (13)
' R Rw
" Boundar_Conditions
Demtray and Levtnsonconsidered the case of the bag resting on a flat surface. In
this work, we consider the membraneacted uponby loads uniformly distributed on a
line which is perpendicular to the (x,y) plane. In the first case, we consider a
central11ne loadas shownin Fig 3. The perimeterof themembranehas a total
length2_R _md themembraneis filled_ith a buoyantfluid. The loadintensityis
Q perunit length. TakingQ/R2was the dimensionlessloadper unitaxiallengthof
bag and settingthisequalto th_ buoyancyforceper unit lengthwe obtain
Q . _" dA
_', )A _- (14)
The equllibrlumequationat tl_e pointof appllcatlonof the forcei.e.at sir = e Is
_F(QIR ,)n]e, arc stn(q/ZT) , arc sin U T/poR)(p°/Rw) (15)
in terns of the abovedimensionless groups.
2O2
I/
i
1976007927-209
Equations(6),(7)and(15)may be solvedsimultaneouslyon a digitalcomputerby "_
an tterattve process. -_+---- -x,
Q
Y F
u ,
f
Fig 3 Central SuspensionCase Fig 4 Twin SuspensionCase
In the secondcase, we consider two equal line loads symmetrically disposed about
i the centreline as shownin Fig 4. The force intensity F arises ph)stcally from a
; sat of tnextenstble cables which pass through sliding seals at the bottom of the
membraneand anchor to a rigid frame of width 2e. for convenience it is assumed
that if the membranewere circular, the lower anchor point would be in the plane
y = O; under equilibrium conditions with tim membranedeformed the anchor point has
,t fallen a distance h. The angle of inclination of the suspensioncables in the
/: undeformedstate is designated by y. In Fig 4, variables with the subscript 1 refer
"t
to the lower portion of the membranebelow the point of application of the load;
: variables with the subscript 2 refer to the upper section. For the upper section
axis Y2 is directed downwardsas sho_ and w is nownegative. The differential
pressure at Y2 " 0 is po - wY.
The boundary conditions arising from continuity of the membraneare
Xlc /R - x2c /R , ylc /R + yzc /R - Y/R
(16)
Sic /R + s2c /R-
The equilibrium equation at the point C yields the further condition
" (Xlc /R)-(e/R) -(T2/poR)cos e_c - (Tl/p,R)cos elc
(17)
(ylc /R)+(h/R) (T2/PoR)stn Bzc - (Tl/poR)sin BIC )
where e/R is the dimensionless cab frame width and h/R t_ the dimensionless l#
suspension deflection.
?
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1976007927-210
- " • {// •II
• _._ t+
+ Ii, NUMERICALRESULTS '
Central Support Case t:
-* Membraneshapesand stress resultants were obtained for typical conditions which
_ + ' '+" mightapplyto a smallnon-rlgidairship,i.e.diameter= 40 ft, inflationpressure
.+
++_ in the range5 to 15 Ibf/ft2 and w = 0.0696Ibf/ft3 which is the 11ftof pure
> hell_nat 0% (thisis a conservativevalue,0.0625lbf/ft3 oftenbeingtakenfor
airshipcalculations(5)). The resultsare presente,l withdimenszonsfor convenience.
l
+_ Fig 5 showsthe deflectedshapesand TableI the stressresultantsand deflections
of the suspensionpointfromthe positionfor a circ_qarmembrane. i
:. __] Table I I
_,,.._.-- po(Ibf/ft2) T(lbf/ft2) h(ft) q(Ibf/ft) I
,- 5 112.3 5.233 85.6 I _
10 212.9 2.727 86.6 I
+ 15 313.3 1.831 87.0
_.- UN_rL[CT(D
"* _/'-_ • 5 IDe/_*; TED
30
" i;-, ]Q
,, + / ]
•,+ / ]i++ i_
k a' _ " W) N
_ZON?A/. (]_TANC£, • (I rO_ZI_TAL []6TAI_(, •
Fig 5 Cross-sectional Shapes Fig 6 Cross-sectional Shape
for various Inflation pressures for an Inflatlonpressure of
for the central suspensioncase 10 lbf/ft 2 for twin suspension
TwoSupport Case
This ca_e, show_ in F_g4, is the more usual situation tn atrshtps and is considered
further. Results were obtaiaed for the particular geometry _ - 15° and e/R - 0.2.
Ftg 6 sho_s the deflected shapefor a membraneof nominal diameter 40 ft andan
Inflation pressure po - 10 ]hf/ft 2. Fig 7 showsthe non-dimensional deflection h/R
versus the non-dimensional pressure parameter po/l_. Fig 8 showsthe non-dtr_nstonal
membranestresses T/poR versus po/l_.
l
1976007927-211
1
INTERNAL_SURE FA,_IMb'T_, _/_ _ERNAL m£_IJ_E i_Ai_r1'ER, _/_
Fig 7 Suspension Deflection, Fig 8 Stress Resulatants, T/poR r
h/R, versus po/Rw for the case versus po/l_ for the case shown tn
shown tn Ftg 4; e/R-0.2, y-15 ° Ftg 4; e/R-0.2, y-15 °
THE DYN_IC CASE
He consider the case tn which the bag and the atr surrounding it are subject to a
verttcal acceleration ng upwards. If we assume that the differential pressure at
y - 0 ts Po as before, tt may eastly be shown that for the dynamic case, the
differential pressure at any othe_ potnt ts
p' - po+ (n+l),y (18)
The buoyancy force per unit length will be
_'_ Q' - w(n+l) _A dA (19)
We consider that the suspension ts attached to a mass which under static conditions
gives rlse to a vertical force per unit lengthQ-W/AdA (from equation 14). Under
dynamic conditionsthe vertical componentof the suspensionforce will now be Q(l+n)
and clearly this Is equal to Q'. Thus the dynamic case may be obtained from the
preceedlng results by replacingthe relative specificweight w by w' where
w° - (l+n)w {20)
DISCUSSION
TI;e analysis provides membrane shapes and stress resultants for the two-dimensional
problem of the fluid ftlled bag and these can be applled to both the static and
dynamic cases.
In a non-rigid atrsiltp, the inflation pressure po must be sufficient to mat,tatn the
shape of the structure under both static and dynamic loads. As an example, _e
consider an airship designed for a 75 mph maximumspeed. Allowing for a frontal
2O5
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1976007927-212
-T ', o
r
• e
" : gust of 15 mph, the maximumstagnation pressure at the nose Weuld be 22 lbf/ft 2.
"' Usually the nose contains a stiffening structure, which pemtts lower inflation
pressures of, for example, 60% of the stagnation pressure or 13 lbf/ft 2 tn thts case.
_. For a helium filled bag of 40 ft diameter, the non-dimensional pressure parameter
_, would have the value of 9.3. From Ftgs 7 and 8, we obtain values for the verttcal
t
deflection of 8.4 inches and for the stress resultants of TI" 270 lbf/ft and
_ _ Tz" 292 lbf/ft. The stress resultant due to pressure only, t.e. poRts 260 lbf/ft
so the effect of the suspension on the envelope stresses ts quite small.
In the design of airships, tt Is customary to consider the effect of a transverse "
I
gust of about 30 ft/sec (7). A vertical gust of this magnitude could give rtse to
, accelerations tn excess of lg. In the example chosen, the parameter po/Rw would
,.._- be halved for an upward acceleration of lg and the deflection would be 16.6 inches ,
t.e. approxtmtely doubled.
There are important differences, however, between the problem fomulated and the
real case of an airship. These are:-
1. The analysis ts for a two dimensional system. Airships wtl1 have
a fineness ratio (overall length to maximumdiameter) of between
3 and 5, thus curvature tn the axial plane wlll significantly diminish
"' the stress resultants and probably increase the overall stiffness.
_: 2. It ts not possible to arrange the suspension system in such a way
_: that the suspension force F tn Ftg 4 exactly balances the buoyancy
at that section. Consequently bending moments arise In the axial
plane and the assoclated shear forces are transmttte_ through the
membrane. These wtll glve rtse to deformations of the section which
differ from those tn the two-dimensional case.
3. It ts customary to have a
secondary suspension system
tn the form of a skirt or
fairing between the cab and
the envelope as shown tn the
schematic diagram tn Fig 9.
Thts wtll be considerably
stiffer than the upper sus-
pension system so that under
dynamic loadtng the addi-
tional loads wtll be trams- Ftg g Schematic Illustration
ferred to the envelope by shoving the skirt location
3O6
1976007927-213
J 41
¢',
!
the skirt rather than by the cables.
4. Below the cetltng altttude of the atrshtp, not a11 of the envelope
ts ftlled wtth helium. Up to 10_ of the internal volume can be taken
up by air tn ballonets as shown tn Ft9 1. These serve to maintain
the Inflation press_tre and allow for expansion of heltum at higher J
altitudes t.e. on ascending atr ts bled off from the ballonets and
thts prevents loss of heltum. These ballonets may have a significant
effect on the. deformed shape at a sectton.
5. The dynamtc case assumes that the _urroundtng flutd has the same
l
acceleration as the bag. Thts is not truly representative of the
situation in a verttcal gust where there wtll be an aerodynamic
pressu_'e distribution on tP_ section due to the relative transverse r
veloctty of the surrounding atr. Thts proble,, as well as the effect
of the pressure distribution due to forward veloctty are outside
the scope of thts work.
Other factors gtve rise to stress distributions and deformations in airship
envelopes whtch have not been considered here. These tncl_de instantaneous and
creep strains tn the fabric, t,lproper rigging and the effects of the empennage.
]t ts considered, however, that the analysts and numerical results presenLed wt]l
assist th_ designer tn the preliminary investigation of envelope and suspension
performance tn non-rigid airships.
ACKNOWLEDGEHENTS
The authors would ltP:e to thank the Nattonal Research Counctl for their support
of the research at lldlaster University and Hr. R. Schneider N.A., President of
the Canadian Airship Development Corporation and Hr. E. DeAssis and Hr. D. Epp;
of HoverJet ]nc._ for their help and advice tn ti;Is work.
)
RErERENCE$:
1. H.Oemtray and H. Levtnson, Int. a. Hech. Set. 14, I)431 (1972).
Z. F. Otto, Tensile Structures Vol. 1, HIT Press, Cambrtdge,Hass.(1967).
3. 6.H. _, a. Appi. Hech. 71, p_:5 (1971).
4. J.L. Duncan and l..R.Ragal_, Trans CSHE1, p135 (1972).
207
1976007927-214
i
i
t 5. C.P. Burgess, Airship Ueston, Ronald Press, New York (1927).
6. I.S. Gradshteyn and I.H. Ryzhtk, Table of Intec)rals Series and Products,
Acad_mtc Press, New York (1%5).
J
" ', 7. H|ntsgry of Transport, Civil Aeronautics _ovtstonal Airworthiness
: Requirements, Airships (Canadian Government) (derived from FAR.25)
/
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1976007927-215


Long fluid filled bags suspended by line forces

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