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Spherical membranes subjected to concentrated loads
Spherical membranes subjected to concentrated loads W. Szyszkowski and P. G. Glockner Department of Mechanical Engineering, The University of Calgary, 2500, University Drive NW, Calgary, Alberta, Canada T2N IN4 (Received April 1986)
Using linear membrane theory, the contours and size of reinforcements required for spherical inflatable membranes to carry tangential, normal and oblique concentrated forces, by means of a tension field and without wrinkling, is considered. Load limits for such wrinkle-free behaviour of this class of structure are established. In the second part of the paper, nonlinear membrane equations, written with respect to the deformed configuration, are used to establish the wrinkled deformation field of spherical inflatables subjected to axisymmetric normal concentrated forces without the use of reinforcements. The ponding problem for such membranes, a possible mode of failure when concentrated forces are applied in the presence of an accumulating ponding medium, is defined and the conditions for a safe ponding process specified in terms of membrane geometry, internal inflation pressure and density of the ponding medium. Keywords: spherical membranes, tension field, wrinkling, reinforcement, ponding With continuing improvements in the mechanical/ strength properties of commercially available plastic sheets, the use of single-walled inflatables for the creation of large-scale enclosures, primarily of cylindrical and spherical shape in agriculture, exploration and industrial applications, as well as for sports and recreational facilities, continues to increase. ~.2This continued interest in the use of pneumatic structures is due, in part, to their relative economy and ease of erection, as compared to structures made of more traditional materials. When dealing with such inflatables, due to the total lack of bending and compressive stiffness of the membrane, application of concentrated loads should, in general, be avoided. In practice, however, such concentrated loads are sometimes unavoidable. One should then know how to design for such loads and, in particular, try to eliminate their undesirable effects, including tearing, wrinkling and/or unacceptably large deflections. Such deflections may lead to ponding instability in the presence of snow, rain and/or ice ?,4 Concentrated or line loads are the result of certain functional demands which require the hanging of equip0141-0296/87/01045-08/$03.00 © 1987Butterworth& Co (Publishers) Ltd
ment from the roof, resulting in forces with components both normal and tangential to the membrane (Figure 1(b)). High-profiled spherical inflatables used in military applications and in research for protection of highly sensitive instruments, have to be stabilized against wind by cable tie-downs and guy-wires. Such cables or wires, when attached to the inflatable, result in tangential forces which have to be transmitted into the membrane 'safely', without causing undue distortions and/or rupture or wrinkling (Figure l(a)). In the first part of the paper, spherical inflatables subjected to tangential, normal and oblique concentrated loads are analysed. Linear membrane theory is used to determine the size and contour of stiff reinforcing elements required for such forces to be transmitted/distributed into the membrane so as to create a pure tension field without wrinkling. Load limits are established for such wrinkle-free behaviour. The determination of wrinkled deformations for such membranes requires the application of nonlinear membrane theory, in which the equations are written with respect to the deformed state. Large (wrinkled) deformations of spherical membranes subjected to normal Eng. Struct., 1987, Vol. 9, J a n u a r y
45
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner
A
RO
a
"
b
Figure 1 Spherical inflatables subjected to concentrated loads: (a) tangential load applied to high-profile structure; (b) normal load applied to flat m e m b r a n e
concentrated loads applied at the apex are treated in the next section of the paper. Finally, the ponding problem, 3,4 a mode of failure of such membranes when subjected to concentrated loads in the presence of a ponding medium accumulating in the depression caused by such loads, is examined. Ponding instability of spherical inflatables subjected to such loads is analysed in detail, determining the parameters for a safe ponding process, including initial geometry, density of the ponding medium, internal inflation pressure, and load magnitude.
Wrinkle-free behaviour of spherical membranes subjected to concentrated loads Consider a spherical inflatable of radius R(j, central halfangle d~0, subjected to an internal pressure, p,, and various concentrated loads. Figure 1 (a) shows tangential concentrated forces applied to a high-profiled spherical inflatable, resulting from stabilizing tie-down cables. Linear membrane theory is used to analyse this problem. If wrinkling is to be avoided, no negative membrane stresses are allowed, a criterion which helps to define the shape and size of a reinforcement required to prevent wrinkling due to the application of such concentrated loads.
Tangential concentrated loads Introducing the local coordinates. 4, and 0, referenced to the arbitrary tangential load, T, applied at point A. (see Figure 2), the stress resultants N,t,. No and N,bo are found to be: ~
N,~(ch,O) = Nl(~h) cos0 + N. Ne(&,O) = -Nj(4~) cos0 + N~
(lb)
N~o(&,O) = N~(&) sin0
(lc)
(la)
where:
Nm,~, } = N~ + N I
Nmin
(3a)
= 0/2
(3b)
Interestingly, the principal stress resultants are constant along any circle around the point of load application, point A, as shown in Figure 2(d). The circle for which Nmi n = 0 defines the boundary between admissible and inadmissible membrane stress states for wrinkle-free behaviour (Figure 3). Within this circle, Nmi. < () and the membrane must be reinforced, while outside this b o u n d a r y , Nmin > 0, i.e. a tension field exists and no reinforcing elements are required. The angle ~b~ associated with this boundary (at which Nmi, = 0) is defined, for a given load magnitude and geometry, from the equation:
I
T- 5
sinqS~(1 + cos05~) = (I
(4)
where T = T/TrR{~p~. Variation of T with O,. is shown in Figure 4. Equation (4) defines the admissible tangential load magnitude for 'wrinkle-free' behaviour, provided a stiff element is used to reinforce the membrane within the circular area defined by the angle 05~,. Loads greater than this limiting value should result in wrinkling of the membrane simultaneously all around the stiffener and in directions parallel to the major principal stress resultant, N ...... (Figure3(a)). The absolute maximum tangential load level, T = 0 . 6 5 , corresponds to &c=60 °. Stiffeners covering areas with subtended angles larger than 60 ° do not add to the carrying capacity of the membrane.
Normal concentrated loads'
NI(05 ) =
N~, -
stress resultant magnitudes in the principle directions have to be determined. For this purpose, a Mohr's circle construction is used (Figure 2(c)) and the principal stress resultants, N ...... and Nm,, and associated directions determined as:
T ~R~ sin~b( I + cos~b)
l)oRo 2
(2a)
(2b)
To decide whether this membrane stress state meets the stated criterion for 'wrinkle-free" behaviour, the
46 Eng.Struct., 1987, Vol. 9, January
In the case of a concentrated load. W, being applied to the membrane in the surface normal direction, the principal directions coincide with the coordinate directions ,.h and 0 and the principal stress resultant magnitudes are given by: N ..... t Nmin
=- No+ N~
N~ = W/2rrR, sin:&
(5)
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner
N~8 ilN~
T
b
N I cos O * N o
Nmox
Nmin
N¢, N o
N I cos
0+
No
N¢o
Nmin
c
d
Figure 2 Effects of tangential concentrated load on spherical surface: (a) notation--side view; (b) frontal view; (c) Mohr's circle; (d) principal stress state
4
Zone of pure /tension field
/
Clearly, any stiffener should be axisymmetric about the point of load application and the load limit for wrinklefree behaviour is defined in terms of the angle +c by:
W = W/~rpoR~, = sin2d~c // / /
1
T
a
b
Figure 3 Local reinforcement for tangential concentrated load: (a) principal stress state around reinforcement; (b) side view of reinforcement
(6)
This relation is, of course, valid for W being directed inward or outward. The variation of W with d~c is also shown in Figure 4. As opposed to the tangential load case, this curve does not exhibit a maximum for ~c < 90 °. As discussed for the tangential loading, normal force magnitudes in excess of I~, defined by equation (6), result in simultaneous wrinkling all around the stiffener, which happens to be a loading/wrinkling case examined in detail in reference 6.
Oblique concentrated loads In the case of a concentrated load, F, applied to the membrane at an angle cr with the surface normal (Figure 5), the membrane stress resultants are obtained as: N~ = Ni cos0 + N, + N,
Eng. Struct., 1987, Vol. 9, January
(7a)
47
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner
W
" Yc
O O
0.65 L. \
G)
=o
0.5
O o
\
¢)
E z
01¢ 0
I0
20
:50
40
Angle defining
50
60
70
reinforcement
80
90
size,~c(°)
Figure 4 Variation of tangential and normal concentrated load with reinforcement size
y/R
/-
kylR
;~
CIq
F
H T-~ ~ ~ . B
\\
I0
z/t? ~ 1 - - - - ~ - ; ~
DI
~min= 6,17 °
x/R A
I ¢
= 10.2 °
/ a =0 °
: 16.4 °
/
\
/ /./"
E
I0 o
c 1v
a
'
f j
b
Figure5 Local reinforcement for oblique load: (a)side view; (b)frontal view (~ = 10°; # = 0.175)
48
Eng. Struct., 1987, Vol. 9, January
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P, G. Glockner
No = -NI cos0+ N o - N2
(7b)
N+o = -N~ sin0
(7c)
in which the definitions for N0, Nj and N2 remain unchanged. The principal stress resultants and principal directions for this case are defined by:
I. 0
a : 90 ° 85 °
~o o.8 ~ lu.,,
75 °
a=O
NminNmax} =No+__(N~j+N~+2NIN2cosO) L / 2 .
(8a)
60 °
°
~o_ o . 6
45 ° 50 °
tan 2tO =
sin0 cos0 + N ff N 2
(Sb)
Adopting, again, the criterion for wrinkle-free behaviour (Nmi n = 0), one can determine the contour separating the areas of admissible and inadmissible stress states. The load limit, F, is expressed as:
O c O
~ 0.2 O Z
0 ~ 0
= I[ 2cosc~ .]2.1- I since/ P [Lsin6(1 + cos6) [sin26J [. 2cosa ] sinc~ ]-,/2 +2Lsin6(1 + cos6) ] s i ~ C ° S 0 I '
~ 0.4
10
20
i
I
I
I
t
I
i
30
40
50
60
70
80
90
Angle defining reinforcement size,
Figure 6
4%(*)
V a r i a t i o n o f o b l i q u e c o n c e n t r a t e d load w i t h d,c
Concentrated normal loads without reinforcement (9)
where je= F/rrp~R~ and where the relations W = Fsina, and T = F cosa were used in connection with equations (2a), (5) and (8). For given values of/~ and a, equation (9) defines a closed contour in the coordinate system (6, 0) defining the boundary of the reinforcement required for wrinkle-free behaviour. For a = 0 or 90 °, equation (9) results in the relationship 6 = constant, confirming the results presented in the two previous subsections. In general, a surface contour, 6 = 6(0), represents a three-dimensional curve, an example of which is shown in Figure 5 for i f = 0.175 and ~ = 10° (line BDC). For comparison purposes, the circular contour, line EH, corresponding to F = 0.175 and a = 0 ° (tangential load) is also shown in this diagram. As is clear from Figure 5, even a relatively small normal load component results in a significant increase in the size of the required reinforcement. The extrema for 6 are obtained at 0 = 0 ° (6re,x) and 0 = 180° (6rain). Along the line BDC, Nmi n vanishes while the other principal stress resultant along this contour has a constant magnitude, Nm~,x= poR, (a value twice as large as the membrane stress due to internal pressure), with the direction of its projection onto the X - - Y plane being defined by the ray connecting the projection of the point in question along the contour with point B. As can be observed from Figure 5, the boundary of the required reinforcement is quite complex, which may not be convenient a n d / o r economical from a production viewpoint. Consequently, one might prefer to replace such a stiffener with a circular reinforcement defined by the angle 6 ..... and indicated in Figure 5 by the contour C ' D ' C . With such an enlarged stiffener, Nm~nwill vanish only at point C, while at the remainder of the contour Nmi n > 0. Load limits, F, as a function of 6 .... = 6c, and for given values of ce are indicated in Figure 6. Increasing the reinforcement from the obtained contour to a larger circular area does not, of course, increase the value of F, since this value is governed by the Nmi, = 0 condition at point C.
In the previous section a linear membrane theory was used to determine the shape and size of stiffening elements. The application of the linear theory was justified because local reinforcements facilitated the transmission of concentrated loads into the membrane without wrinkling and with the original (spherical) shape remaining intact. In the absence of any such local reinforcements, application of concentrated loads will result in local wrinkling which, in turn, may change the geometry of the structure significantly. Consequently, such deformations can only be determined using nonlinear membrane theory, the equations for which can be written either with respect to the current deformed state (Eulerian description) TM or with respect to the undeformed state (Lagrangean description). 2.9 In the former case, the coordinate system, associated with the current configuration has to be determined while in the material (Lagrangean) formulation, a finite displacement vector is used as a key variable. In the case of normal concentrated loads applied at the apex, the deflection field must be axisymmetric and, therefore, the Eulerian approach is more convenient.
Governing equations for the wrinkled region Assuming inextensibility, the membrane surface can be divided into wrinkled and undeformed regions (Figure 7(a)). In the wrinkled domain, the hoop membrane stress resultant, No, must vanish and the equations of equilibrium, written with respect to the deflected/ wrinkled configuration, have the form: 5 d
- - ( R d V 6 sin&) = 0 dO N&
R,~
-p
(lOa)
(10b)
where p is the normal pressure, in general varying along the meridian, and R e and Ro denote radii of curvatures in the circumferential and meridonial directions, respectively, defined for an axisymmetric surface by:
Eng. Struct., 1987, Vo1.9, January 49
Spherical m e m b r a n e s subjected to concentrated loads: W. Szyszkowski and P. G. Glockner I Wrinkled regmn 1-4~
"/'4. LF
rW !
/
Wrinkled region
h
a
Undeformed region
b
Unreinforced spherical membrane subjected to axisymmetric concentrated load: (a) qualitative representation of problem; (b) cross-section of deformed profile Figure 7
r
R,-
(lla)
sind)
R6-
05
"11"I- ~7"
dr - cos4) d& 1
(lib)
and where the coordinate r, as well as h, is shown in Figure 7(b). From equations (10) and (11) one obtains:
,g ~°
0.4
C o n c ~ / / / load only ~//
/
dr C -- =--cos(b
(12)
dcb rp
where C is a constant of integration. For constant pressure, p = Po, equation (12) can be solved directly. Using the boundary conditions r ( - - a A ) = 0 and r(&L)= &, sin&L, one obtains:
I~
o3
5) ;~L---~ 5"
_
/
~0=o.235~/
-~ -~ -g g
; e/
2
i, /
ntr t
G
/ s i n ( b + s i n a a )j'2 sin&l/---" \ s i n & L + sinaA
r=&,
(13)
Replacing the variable (b in equation (12) by h gives: d2hdr 2 2"n'rsinc~a [ W p l+\(dh]2]~2=0dr!
z -
-7/7
O
E
rF <--_r<_%r[.
(15)
equation (14) is solved numerically for two separate domains, namely r < r[- and r > rE and subjected to the boundary conditions: h(0) = h A"
dh dr],.=,,
-
h(rt:) = 0;
O:
dh dr1,=,.
dh drl,=.
-
tan~a;
- tan&l,
(16)
The solution of equation (14) leads to two arbitrary integration constants which, together with the parameters hA, rE, aA and &t. appearing in equation (16), constitute the set of six unknowns to be determined. There are five boundary conditions given by equation
50
Eng. Struct., 1987, Vol. 9, January
_
)"Ro
Y -
Po
;=5
8c = I 062 0
{Po
/~
0.1
(14)
an equation which is useful when considering the ponding problem in which the depression caused by the concentrated load is filled with some ponding medium of density % For such cases, the pressure p is given by:
P= Po
I d
g
0
[ 0.5
I0
I 1,5
Nondimensionol deflection,5 = 8/R o Figure 8
Load-deflection curves for spherical membrane
(16); the necessary sixth equation is obtained from compatibility considerations, considering inextensibility of the membrane in the meridional direction, defined by: si,,4 dr ii cos& - R()&L
f
(17)
Having determined the geometric parameters h A , r E, c~A and (bL, the total deflection of the point of load application is found from: (6, 8 = Ro(1 - cos&L) tan& d r + h a (18) - 0
The system of highly nonlinear equations, equations (14), (16) and (17) was solved numerically using an iterative procedure discussed in reference 10.
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner
Results--ponding instability Typical load deflection curves for the axisymmetrically loaded spherical inflatable are shown in Figure 8 where the dashed line indicates the response of the membrane when subjected to only a concentrated load applied at the apex, while the solid curve depicts the load-deflection behaviour of this inflatable when subjected to an axisymmetric concentrated load in the presence of an accumulating ponding medium. As can be seen from these results, the structure may exhibit limit point behaviour, 7even when subjected only to a concentrated load. Such limit point instability, however, occurs only for relatively high-profiled structures (oh0> 90 °) and at deflection magnitudes exceeding the original radius of the membrane. The effect of an accumulating ponding medium (rain or snow) on the load-deflection behaviour of such a structure, subjected to a concentrated load, W, applied at the apex, is indicated by the horizontal line starting at the dashed curve and terminating at the solid curve, with the former (curve) corresponding to the case of zero density (3' = 0). If such a horizontal line intersects the solid curve, as for example, the line 1'1, a case for which the depression is completely filled, the ponding process is considered to be stable. If, on the other hand, the ponding medium only enlarges the depression, but never fills it completely (see line 5'5"), the configuration and the ponding process are clearly unstable, leading to larger and larger deflections and finally to dynamic collapse. The critical configuration and load magnitude are associated with the accumulation process indicated by the line 2'2 in Figure 8 for which the concentrated load, Wp, is referred to as the critical ponding load. The decreasing portion of the load-deflection curve between points 2 and 4 has a physical meaning only in the case of a 'passive' load caused by tie-downs. '~ In such cases, the distance 3'3 denotes the release in tension in the tie-down cable due to the accumulating ponding medium. Clearly, the largest decrease in tension is defined by the distance 4'4, with the associated critical initial deflection/imperfection, 6~, denoting the largest initial deviation from the perfect shape which can be tolerated without the designer having to be concerned about ponding instability failure. The magnitudes of critical initial imperfections/deflections were discussed in detail in reference 8. The magnitude of the critical ponding load, Wp, is very sensitive to the density of the ponding medium and magnitude of the internal overpressure, as well as the initial geometry as indicated by the initial radius of curvature of the spherical membrane. Variation in the dimensionless load, Wp, as a function of the dimensionless density, ~, is shown in Figure 9. As can be seen from this figure, this relation can be approximated by a straight line in the density range, 20,~ ~/~ 1000, and, therefore, the following approximate design expression can be written: Wp --- 1.1 1 ~1-0.')~,41 Wp~ 3.43
p[/9~,4 RII( 03~, 3'l~.,x,(,
(19)
Experimental results 3 generally indicate critical ponding load values slightly smaller than corresponding theoretical predictions. Interestingly, these results also
%o
..
o
_~
0,0
E_
0.001
\.,.
~ _ i lly(-0.964)
°
"
0.0001
"
I
I
I
1
I0
I00
I000
I0 000
Nondimensional density, ~= ¥Ro l Po L o g - l o g plot of nondimensional load against nondimensional density: ((©) R = O.8m; (+) R = 1.43 m) 3 Figure 9
appear to fall along a straight line on the logarithmic plot shown in Figure 9, thereby confirming, in some sense, the validity of the theory discussed here and its underlying assumptions. The difference between experimental and analytical results can be reduced, at least in part, by taking into account the actual imperfections existing in the structure?
Conclusions Using linear membrane theory, the shape and size of rigid reinforcements for wrinkle-free behaviour in spherical pneumatic structures is determined. In particular, reinforcements for tangential and normal concentrated loads applied to such membranes are established showing that such reinforcements must be circular. For oblique loads the shape of the reinforcement is more complex. The actual contour can, however, be replaced by a slightly larger circular shape with the same load carrying capacity, should such replacement be desirable for economic reasons. When concentrated loads are applied to membrane structures without reinforcement, wrinkling will occur resulting in significant changes in the geometry of the structure. The analysis, therefore, has to be based on nonlinear membrane theory, the equations for which are written with respect to the deformed state. The specific case of a normal concentrated force applied at the apex of a spherical membrane is treated in detail in the second part of the paper. Finally, the ponding problem, in which a ponding medium accumulates in the depression caused by the concentrated load, is considered. The analysis indicates that, even under axisymmetric concentrated loads, the behaviour of spherical inflatables exhibits limit point characteristics, with the magnitude of the critical load being very sensitive to the inflation pressure, the radius of curvature of the membrane and the density of the ponding medium. Results are presented in the form of nondimensional load deflection curves both for the concentrated load only and the ponding problem. These results, and particularly those
Eng. Struct., 1987, Vol. 9, January 51
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner from the ponding analysis, should be of interest to designers of inflatable membrane structures with large spans for which the critical concentrated loads that can be applied safely could be relatively small. Acknowledgement The results presented here were obtained in the course of research sponsored by the Natural Sciences and Engineering Research Council of Canada, grant A-2736. References l 2 3
52
Mead, B. ~Market needs for permanent fabric structures', TDe Design Process, Proc. Int. Syrup, Archit. Fabric Slruct,, Orlando, Florida, 1984, pp 19-22 Frei, O. ~Tension structures', Praeger Publ. Inc., NY, 1970 Malcolm, D. J. and GIockner, P. G. 'Collapse by ponding of air-supported spherical caps', Proc. ASCE, 1981, 107 (ST9), 1731-1742
Eng. Struct., 1987, Vol. 9, January
-J~ S]iva~ta~a. N. K . llanda, \ . K and ('rilchk'~. ~. }',lilmc~ ol air-supportcd slructulc,,'. (117./.,IA'S %'rap lit %zq~p l'7"ames. Proc. 1ASS Pac. S3',lp., Tokw~ and Kvoto. ICJ71. pp 44c)-40(I 7 Szyszkowski. W. and (ilockner. P. (}. "Finilc delormalion and stability hehaviour of spherical inflatables under axisymmelric concentrated loads'. Int. J. n(mlmear Mech. 1984, 19(5), 489-496 8 Szyszkowski, W. and Glockncr, P. G. "Finite deformation and stability behaviour of spherical inflatables subicc[cd to ~lXiSVmmetric hydrostatic loading', Ira. ,1. Solirts Strftcl.. 1t)S4, 2(I ( I 1' 12), 1021-1(t3(~ t) Mcgara, W,. O k e m u r a , K. and Kawaguchi, M. A n analysis ol mcnlbranc slructures', Shell anU Slmtial Struct, Enq. Proc. lASS ,S~w~/p., 1983, Rio de Juneiro, pp 1-12 10 Glockncr, P. G. and Szvszkowski, W. 'Some stability considerations of inflatable structures', Shell apld SpamU .S'trucl. in Espy. Proc. lASS Strop.. Rio de Janeiro. 1983. pp 11f>134
Using linear membrane theory, the contours and size of reinforcements required for spherical inflatable membranes to carry tangential, normal and oblique concentrated forces, by means of a tension field and without wrinkling, is considered. Load limits for such wrinkle-free behaviour of this class of structure are established. In the second part of the paper, nonlinear membrane equations, written with respect to the deformed configuration, are used to establish the wrinkled deformation field of spherical inflatables subjected to axisymmetric normal concentrated forces without the use of reinforcements. The ponding problem for such membranes, a possible mode of failure when concentrated forces are applied in the presence of an accumulating ponding medium, is defined and the conditions for a safe ponding process specified in terms of membrane geometry, internal inflation pressure and density of the ponding medium. Keywords: spherical membranes, tension field, wrinkling, reinforcement, ponding With continuing improvements in the mechanical/ strength properties of commercially available plastic sheets, the use of single-walled inflatables for the creation of large-scale enclosures, primarily of cylindrical and spherical shape in agriculture, exploration and industrial applications, as well as for sports and recreational facilities, continues to increase. ~.2This continued interest in the use of pneumatic structures is due, in part, to their relative economy and ease of erection, as compared to structures made of more traditional materials. When dealing with such inflatables, due to the total lack of bending and compressive stiffness of the membrane, application of concentrated loads should, in general, be avoided. In practice, however, such concentrated loads are sometimes unavoidable. One should then know how to design for such loads and, in particular, try to eliminate their undesirable effects, including tearing, wrinkling and/or unacceptably large deflections. Such deflections may lead to ponding instability in the presence of snow, rain and/or ice ?,4 Concentrated or line loads are the result of certain functional demands which require the hanging of equip0141-0296/87/01045-08/$03.00 © 1987Butterworth& Co (Publishers) Ltd
ment from the roof, resulting in forces with components both normal and tangential to the membrane (Figure 1(b)). High-profiled spherical inflatables used in military applications and in research for protection of highly sensitive instruments, have to be stabilized against wind by cable tie-downs and guy-wires. Such cables or wires, when attached to the inflatable, result in tangential forces which have to be transmitted into the membrane 'safely', without causing undue distortions and/or rupture or wrinkling (Figure l(a)). In the first part of the paper, spherical inflatables subjected to tangential, normal and oblique concentrated loads are analysed. Linear membrane theory is used to determine the size and contour of stiff reinforcing elements required for such forces to be transmitted/distributed into the membrane so as to create a pure tension field without wrinkling. Load limits are established for such wrinkle-free behaviour. The determination of wrinkled deformations for such membranes requires the application of nonlinear membrane theory, in which the equations are written with respect to the deformed state. Large (wrinkled) deformations of spherical membranes subjected to normal Eng. Struct., 1987, Vol. 9, J a n u a r y
45
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner
A
RO
a
"
b
Figure 1 Spherical inflatables subjected to concentrated loads: (a) tangential load applied to high-profile structure; (b) normal load applied to flat m e m b r a n e
concentrated loads applied at the apex are treated in the next section of the paper. Finally, the ponding problem, 3,4 a mode of failure of such membranes when subjected to concentrated loads in the presence of a ponding medium accumulating in the depression caused by such loads, is examined. Ponding instability of spherical inflatables subjected to such loads is analysed in detail, determining the parameters for a safe ponding process, including initial geometry, density of the ponding medium, internal inflation pressure, and load magnitude.
Wrinkle-free behaviour of spherical membranes subjected to concentrated loads Consider a spherical inflatable of radius R(j, central halfangle d~0, subjected to an internal pressure, p,, and various concentrated loads. Figure 1 (a) shows tangential concentrated forces applied to a high-profiled spherical inflatable, resulting from stabilizing tie-down cables. Linear membrane theory is used to analyse this problem. If wrinkling is to be avoided, no negative membrane stresses are allowed, a criterion which helps to define the shape and size of a reinforcement required to prevent wrinkling due to the application of such concentrated loads.
Tangential concentrated loads Introducing the local coordinates. 4, and 0, referenced to the arbitrary tangential load, T, applied at point A. (see Figure 2), the stress resultants N,t,. No and N,bo are found to be: ~
N,~(ch,O) = Nl(~h) cos0 + N. Ne(&,O) = -Nj(4~) cos0 + N~
(lb)
N~o(&,O) = N~(&) sin0
(lc)
(la)
where:
Nm,~, } = N~ + N I
Nmin
(3a)
= 0/2
(3b)
Interestingly, the principal stress resultants are constant along any circle around the point of load application, point A, as shown in Figure 2(d). The circle for which Nmi n = 0 defines the boundary between admissible and inadmissible membrane stress states for wrinkle-free behaviour (Figure 3). Within this circle, Nmi. < () and the membrane must be reinforced, while outside this b o u n d a r y , Nmin > 0, i.e. a tension field exists and no reinforcing elements are required. The angle ~b~ associated with this boundary (at which Nmi, = 0) is defined, for a given load magnitude and geometry, from the equation:
I
T- 5
sinqS~(1 + cos05~) = (I
(4)
where T = T/TrR{~p~. Variation of T with O,. is shown in Figure 4. Equation (4) defines the admissible tangential load magnitude for 'wrinkle-free' behaviour, provided a stiff element is used to reinforce the membrane within the circular area defined by the angle 05~,. Loads greater than this limiting value should result in wrinkling of the membrane simultaneously all around the stiffener and in directions parallel to the major principal stress resultant, N ...... (Figure3(a)). The absolute maximum tangential load level, T = 0 . 6 5 , corresponds to &c=60 °. Stiffeners covering areas with subtended angles larger than 60 ° do not add to the carrying capacity of the membrane.
Normal concentrated loads'
NI(05 ) =
N~, -
stress resultant magnitudes in the principle directions have to be determined. For this purpose, a Mohr's circle construction is used (Figure 2(c)) and the principal stress resultants, N ...... and Nm,, and associated directions determined as:
T ~R~ sin~b( I + cos~b)
l)oRo 2
(2a)
(2b)
To decide whether this membrane stress state meets the stated criterion for 'wrinkle-free" behaviour, the
46 Eng.Struct., 1987, Vol. 9, January
In the case of a concentrated load. W, being applied to the membrane in the surface normal direction, the principal directions coincide with the coordinate directions ,.h and 0 and the principal stress resultant magnitudes are given by: N ..... t Nmin
=- No+ N~
N~ = W/2rrR, sin:&
(5)
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner
N~8 ilN~
T
b
N I cos O * N o
Nmox
Nmin
N¢, N o
N I cos
0+
No
N¢o
Nmin
c
d
Figure 2 Effects of tangential concentrated load on spherical surface: (a) notation--side view; (b) frontal view; (c) Mohr's circle; (d) principal stress state
4
Zone of pure /tension field
/
Clearly, any stiffener should be axisymmetric about the point of load application and the load limit for wrinklefree behaviour is defined in terms of the angle +c by:
W = W/~rpoR~, = sin2d~c // / /
1
T
a
b
Figure 3 Local reinforcement for tangential concentrated load: (a) principal stress state around reinforcement; (b) side view of reinforcement
(6)
This relation is, of course, valid for W being directed inward or outward. The variation of W with d~c is also shown in Figure 4. As opposed to the tangential load case, this curve does not exhibit a maximum for ~c < 90 °. As discussed for the tangential loading, normal force magnitudes in excess of I~, defined by equation (6), result in simultaneous wrinkling all around the stiffener, which happens to be a loading/wrinkling case examined in detail in reference 6.
Oblique concentrated loads In the case of a concentrated load, F, applied to the membrane at an angle cr with the surface normal (Figure 5), the membrane stress resultants are obtained as: N~ = Ni cos0 + N, + N,
Eng. Struct., 1987, Vol. 9, January
(7a)
47
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner
W
" Yc
O O
0.65 L. \
G)
=o
0.5
O o
\
¢)
E z
01¢ 0
I0
20
:50
40
Angle defining
50
60
70
reinforcement
80
90
size,~c(°)
Figure 4 Variation of tangential and normal concentrated load with reinforcement size
y/R
/-
kylR
;~
CIq
F
H T-~ ~ ~ . B
\\
I0
z/t? ~ 1 - - - - ~ - ; ~
DI
~min= 6,17 °
x/R A
I ¢
= 10.2 °
/ a =0 °
: 16.4 °
/
\
/ /./"
E
I0 o
c 1v
a
'
f j
b
Figure5 Local reinforcement for oblique load: (a)side view; (b)frontal view (~ = 10°; # = 0.175)
48
Eng. Struct., 1987, Vol. 9, January
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P, G. Glockner
No = -NI cos0+ N o - N2
(7b)
N+o = -N~ sin0
(7c)
in which the definitions for N0, Nj and N2 remain unchanged. The principal stress resultants and principal directions for this case are defined by:
I. 0
a : 90 ° 85 °
~o o.8 ~ lu.,,
75 °
a=O
NminNmax} =No+__(N~j+N~+2NIN2cosO) L / 2 .
(8a)
60 °
°
~o_ o . 6
45 ° 50 °
tan 2tO =
sin0 cos0 + N ff N 2
(Sb)
Adopting, again, the criterion for wrinkle-free behaviour (Nmi n = 0), one can determine the contour separating the areas of admissible and inadmissible stress states. The load limit, F, is expressed as:
O c O
~ 0.2 O Z
0 ~ 0
= I[ 2cosc~ .]2.1- I since/ P [Lsin6(1 + cos6) [sin26J [. 2cosa ] sinc~ ]-,/2 +2Lsin6(1 + cos6) ] s i ~ C ° S 0 I '
~ 0.4
10
20
i
I
I
I
t
I
i
30
40
50
60
70
80
90
Angle defining reinforcement size,
Figure 6
4%(*)
V a r i a t i o n o f o b l i q u e c o n c e n t r a t e d load w i t h d,c
Concentrated normal loads without reinforcement (9)
where je= F/rrp~R~ and where the relations W = Fsina, and T = F cosa were used in connection with equations (2a), (5) and (8). For given values of/~ and a, equation (9) defines a closed contour in the coordinate system (6, 0) defining the boundary of the reinforcement required for wrinkle-free behaviour. For a = 0 or 90 °, equation (9) results in the relationship 6 = constant, confirming the results presented in the two previous subsections. In general, a surface contour, 6 = 6(0), represents a three-dimensional curve, an example of which is shown in Figure 5 for i f = 0.175 and ~ = 10° (line BDC). For comparison purposes, the circular contour, line EH, corresponding to F = 0.175 and a = 0 ° (tangential load) is also shown in this diagram. As is clear from Figure 5, even a relatively small normal load component results in a significant increase in the size of the required reinforcement. The extrema for 6 are obtained at 0 = 0 ° (6re,x) and 0 = 180° (6rain). Along the line BDC, Nmi n vanishes while the other principal stress resultant along this contour has a constant magnitude, Nm~,x= poR, (a value twice as large as the membrane stress due to internal pressure), with the direction of its projection onto the X - - Y plane being defined by the ray connecting the projection of the point in question along the contour with point B. As can be observed from Figure 5, the boundary of the required reinforcement is quite complex, which may not be convenient a n d / o r economical from a production viewpoint. Consequently, one might prefer to replace such a stiffener with a circular reinforcement defined by the angle 6 ..... and indicated in Figure 5 by the contour C ' D ' C . With such an enlarged stiffener, Nm~nwill vanish only at point C, while at the remainder of the contour Nmi n > 0. Load limits, F, as a function of 6 .... = 6c, and for given values of ce are indicated in Figure 6. Increasing the reinforcement from the obtained contour to a larger circular area does not, of course, increase the value of F, since this value is governed by the Nmi, = 0 condition at point C.
In the previous section a linear membrane theory was used to determine the shape and size of stiffening elements. The application of the linear theory was justified because local reinforcements facilitated the transmission of concentrated loads into the membrane without wrinkling and with the original (spherical) shape remaining intact. In the absence of any such local reinforcements, application of concentrated loads will result in local wrinkling which, in turn, may change the geometry of the structure significantly. Consequently, such deformations can only be determined using nonlinear membrane theory, the equations for which can be written either with respect to the current deformed state (Eulerian description) TM or with respect to the undeformed state (Lagrangean description). 2.9 In the former case, the coordinate system, associated with the current configuration has to be determined while in the material (Lagrangean) formulation, a finite displacement vector is used as a key variable. In the case of normal concentrated loads applied at the apex, the deflection field must be axisymmetric and, therefore, the Eulerian approach is more convenient.
Governing equations for the wrinkled region Assuming inextensibility, the membrane surface can be divided into wrinkled and undeformed regions (Figure 7(a)). In the wrinkled domain, the hoop membrane stress resultant, No, must vanish and the equations of equilibrium, written with respect to the deflected/ wrinkled configuration, have the form: 5 d
- - ( R d V 6 sin&) = 0 dO N&
R,~
-p
(lOa)
(10b)
where p is the normal pressure, in general varying along the meridian, and R e and Ro denote radii of curvatures in the circumferential and meridonial directions, respectively, defined for an axisymmetric surface by:
Eng. Struct., 1987, Vo1.9, January 49
Spherical m e m b r a n e s subjected to concentrated loads: W. Szyszkowski and P. G. Glockner I Wrinkled regmn 1-4~
"/'4. LF
rW !
/
Wrinkled region
h
a
Undeformed region
b
Unreinforced spherical membrane subjected to axisymmetric concentrated load: (a) qualitative representation of problem; (b) cross-section of deformed profile Figure 7
r
R,-
(lla)
sind)
R6-
05
"11"I- ~7"
dr - cos4) d& 1
(lib)
and where the coordinate r, as well as h, is shown in Figure 7(b). From equations (10) and (11) one obtains:
,g ~°
0.4
C o n c ~ / / / load only ~//
/
dr C -- =--cos(b
(12)
dcb rp
where C is a constant of integration. For constant pressure, p = Po, equation (12) can be solved directly. Using the boundary conditions r ( - - a A ) = 0 and r(&L)= &, sin&L, one obtains:
I~
o3
5) ;~L---~ 5"
_
/
~0=o.235~/
-~ -~ -g g
; e/
2
i, /
ntr t
G
/ s i n ( b + s i n a a )j'2 sin&l/---" \ s i n & L + sinaA
r=&,
(13)
Replacing the variable (b in equation (12) by h gives: d2hdr 2 2"n'rsinc~a [ W p l+\(dh]2]~2=0dr!
z -
-7/7
O
E
rF <--_r<_%r[.
(15)
equation (14) is solved numerically for two separate domains, namely r < r[- and r > rE and subjected to the boundary conditions: h(0) = h A"
dh dr],.=,,
-
h(rt:) = 0;
O:
dh dr1,=,.
dh drl,=.
-
tan~a;
- tan&l,
(16)
The solution of equation (14) leads to two arbitrary integration constants which, together with the parameters hA, rE, aA and &t. appearing in equation (16), constitute the set of six unknowns to be determined. There are five boundary conditions given by equation
50
Eng. Struct., 1987, Vol. 9, January
_
)"Ro
Y -
Po
;=5
8c = I 062 0
{Po
/~
0.1
(14)
an equation which is useful when considering the ponding problem in which the depression caused by the concentrated load is filled with some ponding medium of density % For such cases, the pressure p is given by:
P= Po
I d
g
0
[ 0.5
I0
I 1,5
Nondimensionol deflection,5 = 8/R o Figure 8
Load-deflection curves for spherical membrane
(16); the necessary sixth equation is obtained from compatibility considerations, considering inextensibility of the membrane in the meridional direction, defined by: si,,4 dr ii cos& - R()&L
f
(17)
Having determined the geometric parameters h A , r E, c~A and (bL, the total deflection of the point of load application is found from: (6, 8 = Ro(1 - cos&L) tan& d r + h a (18) - 0
The system of highly nonlinear equations, equations (14), (16) and (17) was solved numerically using an iterative procedure discussed in reference 10.
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner
Results--ponding instability Typical load deflection curves for the axisymmetrically loaded spherical inflatable are shown in Figure 8 where the dashed line indicates the response of the membrane when subjected to only a concentrated load applied at the apex, while the solid curve depicts the load-deflection behaviour of this inflatable when subjected to an axisymmetric concentrated load in the presence of an accumulating ponding medium. As can be seen from these results, the structure may exhibit limit point behaviour, 7even when subjected only to a concentrated load. Such limit point instability, however, occurs only for relatively high-profiled structures (oh0> 90 °) and at deflection magnitudes exceeding the original radius of the membrane. The effect of an accumulating ponding medium (rain or snow) on the load-deflection behaviour of such a structure, subjected to a concentrated load, W, applied at the apex, is indicated by the horizontal line starting at the dashed curve and terminating at the solid curve, with the former (curve) corresponding to the case of zero density (3' = 0). If such a horizontal line intersects the solid curve, as for example, the line 1'1, a case for which the depression is completely filled, the ponding process is considered to be stable. If, on the other hand, the ponding medium only enlarges the depression, but never fills it completely (see line 5'5"), the configuration and the ponding process are clearly unstable, leading to larger and larger deflections and finally to dynamic collapse. The critical configuration and load magnitude are associated with the accumulation process indicated by the line 2'2 in Figure 8 for which the concentrated load, Wp, is referred to as the critical ponding load. The decreasing portion of the load-deflection curve between points 2 and 4 has a physical meaning only in the case of a 'passive' load caused by tie-downs. '~ In such cases, the distance 3'3 denotes the release in tension in the tie-down cable due to the accumulating ponding medium. Clearly, the largest decrease in tension is defined by the distance 4'4, with the associated critical initial deflection/imperfection, 6~, denoting the largest initial deviation from the perfect shape which can be tolerated without the designer having to be concerned about ponding instability failure. The magnitudes of critical initial imperfections/deflections were discussed in detail in reference 8. The magnitude of the critical ponding load, Wp, is very sensitive to the density of the ponding medium and magnitude of the internal overpressure, as well as the initial geometry as indicated by the initial radius of curvature of the spherical membrane. Variation in the dimensionless load, Wp, as a function of the dimensionless density, ~, is shown in Figure 9. As can be seen from this figure, this relation can be approximated by a straight line in the density range, 20,~ ~/~ 1000, and, therefore, the following approximate design expression can be written: Wp --- 1.1 1 ~1-0.')~,41 Wp~ 3.43
p[/9~,4 RII( 03~, 3'l~.,x,(,
(19)
Experimental results 3 generally indicate critical ponding load values slightly smaller than corresponding theoretical predictions. Interestingly, these results also
%o
..
o
_~
0,0
E_
0.001
\.,.
~ _ i lly(-0.964)
°
"
0.0001
"
I
I
I
1
I0
I00
I000
I0 000
Nondimensional density, ~= ¥Ro l Po L o g - l o g plot of nondimensional load against nondimensional density: ((©) R = O.8m; (+) R = 1.43 m) 3 Figure 9
appear to fall along a straight line on the logarithmic plot shown in Figure 9, thereby confirming, in some sense, the validity of the theory discussed here and its underlying assumptions. The difference between experimental and analytical results can be reduced, at least in part, by taking into account the actual imperfections existing in the structure?
Conclusions Using linear membrane theory, the shape and size of rigid reinforcements for wrinkle-free behaviour in spherical pneumatic structures is determined. In particular, reinforcements for tangential and normal concentrated loads applied to such membranes are established showing that such reinforcements must be circular. For oblique loads the shape of the reinforcement is more complex. The actual contour can, however, be replaced by a slightly larger circular shape with the same load carrying capacity, should such replacement be desirable for economic reasons. When concentrated loads are applied to membrane structures without reinforcement, wrinkling will occur resulting in significant changes in the geometry of the structure. The analysis, therefore, has to be based on nonlinear membrane theory, the equations for which are written with respect to the deformed state. The specific case of a normal concentrated force applied at the apex of a spherical membrane is treated in detail in the second part of the paper. Finally, the ponding problem, in which a ponding medium accumulates in the depression caused by the concentrated load, is considered. The analysis indicates that, even under axisymmetric concentrated loads, the behaviour of spherical inflatables exhibits limit point characteristics, with the magnitude of the critical load being very sensitive to the inflation pressure, the radius of curvature of the membrane and the density of the ponding medium. Results are presented in the form of nondimensional load deflection curves both for the concentrated load only and the ponding problem. These results, and particularly those
Eng. Struct., 1987, Vol. 9, January 51
Spherical membranes subjected to concentrated loads: W. Szyszkowski and P. G. Glockner from the ponding analysis, should be of interest to designers of inflatable membrane structures with large spans for which the critical concentrated loads that can be applied safely could be relatively small. Acknowledgement The results presented here were obtained in the course of research sponsored by the Natural Sciences and Engineering Research Council of Canada, grant A-2736. References l 2 3
52
Mead, B. ~Market needs for permanent fabric structures', TDe Design Process, Proc. Int. Syrup, Archit. Fabric Slruct,, Orlando, Florida, 1984, pp 19-22 Frei, O. ~Tension structures', Praeger Publ. Inc., NY, 1970 Malcolm, D. J. and GIockner, P. G. 'Collapse by ponding of air-supported spherical caps', Proc. ASCE, 1981, 107 (ST9), 1731-1742
Eng. Struct., 1987, Vol. 9, January
-J~ S]iva~ta~a. N. K . llanda, \ . K and ('rilchk'~. ~. }',lilmc~ ol air-supportcd slructulc,,'. (117./.,IA'S %'rap lit %zq~p