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Buckling and vibration of ring-stiffened cones under uniform external pressure
Thin- Walled Structures 6 (1988) 321-342
Buckling and Vibration of Ring-Stiffened Cones under Uniform External Pressure C. T. F. Ross & T. J o h n s Department of Mechanical Engineering, Portsmouth Polytechnic, Anglesca Building, Anglesea Road, Portsmouth POI 3D J, UK (Received 10 July 1987; accepted 4 December 1987)
ABSTRACT Theoretical and experimental results are presented for the buckling and vibration of three ring-stiffened thin-walled cones constructed from mild steel. For vibration, the study consisted of exciting the cones in air and while partially and fully submerged in water. For the buckling study, the cones were subjected to a gradually applied uniform external pressure until instability' occurred.
INTRODUCTION The vibration of ring-stiffened cones under water is of much importance in the design of submarine pressure hulls, off-shore drilling rigs and other similar structures. Such vibrations can be caused by a number of different periodic forces, including those due to the motion of the surrounding fluid and those due to out-of-balance machinery. The modal patterns of ringstiffened cones, due to these periodic forces, are numerous but the present study will be concerned only with the lobar modes of vibration of Fig. 1. The reason for this is partly because these modes correspond to the static buckling modes of ring-stiffened cones under uniform external pressure (Fig. 2), and partly because these modes of vibration are the fundamental modes. It has been found in an investigation on the buckling and vibration of domes ~'5 that, as the external hydrostatic pressure is increased for such vessels, their resonant frequencies decrease, and that as the static buckling pressure is approached the fundamental eigenmodes of vibration become 32l
Thin-Walled Structures 0263-8231/88/$03-50 © 1988 Elsevier Science Publishers ltd. England. Printed in Great Britain
322
L'. 1". F. R,.,~. |
F. J~.,hpt.~ I
l
Fig. 1. Lobar eigenmode of vibration.
Fig. 2. General instability of a ring-stiffened cone.
very similar to the static buckling eigenmodes. Thus it is possible that, when the vibration eigenmodes are of similar form to the static buckling eigenmodes, any small exciting force may trigger off dynamic buckling at a pressure which may be considerably less than the static buckling pressure. Under uniform external pressure, ring-stiffened cones can fail through general instability, where the ring-shell combination buckles bodily, as shown in Fig. 2. The strength of these vessels is dependent on the material
BzLcklittg and vibration of ring-stitCh'tied cymes t~tzder tttr(li)rm external presst~re
323
and geometrical properties of the cone, and two major factors that influence the strength of these vessels are the size and the number of stiffening rings. Initial out-of-circularity of these vessels can also affect their buckling resistances, although this effect is, in general, less serious for long thin shells made from high tensile materials than for shorter and thicker shells made from materials of lower tensile strength. In the present paper, initial out-ofcircularity is defined as the radial departure (plus and minus) from the mean circle. The buckling pressures of near perfect long thin shells constructed from high tensile materials can very often be predicted precisely from elastic instability considerations. For vessels that buckle inelastically, both out-ofcircularity and plastic behaviour of the material of construction must be taken into consideration to determine the magnitude of the elastic knockdown. Elastic knockdown can be described as the difference between the experimentally obtained buckling pressure and the theoretical buckling pressure predicted for a perfect vessel failing through elastic instability. In the near future it is hoped to extend the present investigation to study dynamic buckling, where the modes of vibration can interact with the static buckling mode to cause the vessel to fail at a pressure appreciably less than
Fig. 3. Ring-stiffened cones.
324
{. i /'. h:n~s, l. Jol';:',
that required to cause static buckling. Such an investigation has tflrcad', been successfully carried out on a family of hemi-ellipsoidal domes. ' " The experimental and theoretical investigations of the present study wcrc based on the three machine stiffened thin-~valled cones of Fig. 3, and involved the following: (a) Vibration of the cones in air. (b) Vibration of the cones with water on their external surfaces and air on their internal surfaces. (c) Vibration of the cones fully submerged in water (free-flooded). (d) Vibration of the cones with water on their internal surfaces and air on their external surfaces. (e) Static buckling of the ring-stiffened cones under uniform external pressure.
T H E O R E T I C A L ANALYSIS The truncated conical shell element that was used for this analysis has been described elsewhere, ~ and the ring element was also the same as that previously described? The conical shell element was defined by ring nodes at its ends. as shown in Fig. 4, and it had four degrees of freedom per node, namely, u~, v, w,~and 0,. The ring element had three degrees of freedom, namely, v, ~t~ and O~,and it is shown in Fig. 5, where e is the distance of the centroid of the ringstiffener from the mid-surface of the shell (negative if the frame is external). For both the conical shell element and the ring stiffener element, the w
0
A
~=1 V
ij
Ri
Rj
r
I ~..Axis oflsymmetry oo
t
P a r t section on
A--A
A Fig. 4. Conical element.
Buckling and vibration of ring-stiffened cones under uniform external pressure
I
325
I
Fig. 5. F r a m e with shell.
5
/
tJ
---
6
7q
r
|
2
Fig. 6. C r o s s - s e c t i o n o f a n n u l a r fluid e l e m e n t .
stiffness and mass matrices were based on small deflection elastic theory, but the geometrical stiffness matrices were based on large deflection elastic theory. In o r d e r to represent the motion of the fluid, a 'solid' annular element was used, as shown in Fig. 6. The cross-section of this element was an 8-node isoparametric type, and each node had one pressure degree of freedom. The e l e m e n t is described in greater detail in Refs 1-5. The interaction at the fluid-structure interface was c a r d e d out by coupling the solid and fluid elements at their c o m m o n boundary in the manner described by Zienkiewicz and Newton. t0
EXPERIMENTAL METHOD T h e vessels were machined from a solid block of mild steel, EN1A. Their internal surfaces were machined first, and then each vessel was snugly fitted on to a conical mandrel where its external surface was machined to the nominal g e o m e t r y given in Table 1 and Fig. 7, where N = number of ring-stiffeners b = width of ring-stiffener, of 'rectangular' cross-section
326
c . T. F. Ross, T. Johns
d e R~ R,. h n
= = = = = =
d e p t h o f ring-stiffener, o f ' r e c t a n g u l a r " cross-section ring e c c e n t r i c i t y (negative as the rings were external) r a d i u s o f shell at small e n d = 1.905 cm r a d i u s o f shell at large e n d = 5.08 cm shell t h i c k n e s s = 0.635 m m n u m b e r o f c i r c u m f e r e n t i a l waves o r lobes
The manufacture
of these vessels was extremely good, and plots of the initial
out-of-circularity before and after the vibration tests were carried out are s h o w n in F i g s 8 - 1 0 .
TABLE 1
Geometrical Properties of the Cones M o d e l no.
N
b (ram)
d (ram)
e (ram)
I 2 3
6 6 7
1.016 1.016 1.016
i-016 1.524 2-032
-0.826 - 1.080 - 1.333
t
211.0
r---
)
o
1
I
J
f
J
(:>635 Cone no.
A
B
I 2
I-0 1.0
1.0 1-5
:z
1.o
2.0
Fig. 7. Dimensions (nominal) of the ring-stiffened cones (mm).
Buckling and vibration of ring-stiffened cones under uniform external pressure
(a)
{b)
Befor~ Vibration
After Vibration
Fig. 8. Initial out-of-circularity plots for Cone I.
327
328
C. T. F. Ross, T. Johns
(a)
Befo~ Vibration
( {b)
After Vibration
Fig. 9. Initial out-of-circularity plots for Cone 2.
Buckling and vibration of ring-stiffened cones under uniform external pressure
(a)
(b)
Before Vibration
After Vibration
Fig. 10. Initial out-of-circularity plots for Cone 3.
329
330
C. T. F. Ross, 7". Johns
In Figs 8 and 9 the out-of-circularity plots were taken midway between the third and fourth rings, on the external surfaces of the vessels. In Fig. l0 the out-of-circularity plots were measured at a distance of 11-75 cm from the larger end of the vessel, on its external surface. The importance of taking these out-of-circularity plots was to investigate whether or not the maximum buckled form occurred in the vicinity of the largest initial imperfection. F r o m the plots it can be seen that the vibration tests had little or no effect on improving or worsening the out-of-circularity. It can also be seen that the worst out-of-circularity was that of Cone 1, =0.0127 mm. The initial out-ofcircularity for Cone 2 was =2-5 x 10 -3 mm and that for Cone 3 was =6.4 × 10 -3 mm. The elastic (Young's) modulus (E) was determined experimentally as 1.9 x 10 t' N m -2, and the density (p) and Poisson's ratio (u) were assumed to be 7800 kg m -3 and 0.3 respectively.
VIBRATION Prior to vibrating the cones, initial out-of-circularity plots were taken of all three cones as shown in the upper parts of Figs 8-10. T h e cones were excited with two electromagnetic shakers and with the aid of a frequency response analyser and other associated electronic equipment, as previously described. ,-5 Even lobar modes were obtained by keeping the shakers in-phase, and odd lobar modes were obtained by keeping the shakers 180° out-of-phase. W h e n the cones were vibrated in air, a microp h o n e was used; underwater the microphone was replaced by a hydrophone. For the underwater tests, except for the case of water on the internal surface only, the experiments were carried out in a tank of diameter 1.5 m, as shown in Fig. 11. A f t e r vibrating the cones and before the static buckling tests were c o m m e n c e d , 'initial' out-of-circularity plots were taken again, and these showed that the vibration tests had not affected the initial out-of-circularity appreciably, as shown in the lower parts of Figs 8-10. These figures also show the positions where buckling occurred when the vessels were tested to destruction.
BUCKLING
T h e large diameter end of each cone was bolted to the top of the tank with the aid o f four bolts, and the smaller end of each cone was blocked off by a closure plate, with the aid of another four bolts, as shown in Fig. 12, so it can
Buckling and ~'ibratiott o f ring-stit?ened cr_mes trader tmt torm exterpTal pre'ss"~re
331
Fig. 11. L;ndcr~atcr vibrtltion tests.
be concluded that the larger diameter end was effectively fixed but that the smaller diameter end was "clamped'. T h e r e were four degrees of freedom at each node. namely, an ~xi,l deflection (u°), a radial deflection (w°), a circumferential deflection (~) and a rotational displacement (0), as shown in Fig. 4. In the "clamped" condition. the radial, circumferential and rotational displacements are zero. but the axial displacement is not. To assist in determining the circumferential eigcnmodes due to buckling. each cone had attached to its internal surface ten electrical resistance strain gauges, placed in a circumferential direction, in the region where it xvas expected that the buckled mode would have its largest deflection. Plots of pressure against strain for the strain gauges recording the largest and smallest magnitudes of strain, just prior to buckling, are shown in Figs 13-15. Figures 16-18 show post-buckled forms of the vessels, taken at the positions of the maximum deflection, together with the locations of the strain gauges. These out-of-circularity plots were taken on the outer surfaces of the vessels, where it can be clearly seen that Cone 1 buckled ~vith four circumferential lobes. The out-of-circularity plots for Cones 2 and 3 do not reveal the n u m b e r of circumferential lobes that these two vessels buckled into. F r o m Fig. 13 it can be seen that. just prior to buckling, the measured
332
C I" I-" R,_),~. I J~,~i~:.,
Air w n t
Top plate
Ring
'0' Ring
Ring stiffened cone W~ter
vessel 'O'
Ring
:1 blanking plug
Fig. 12. C o n e in tc~,t tank.
strains suddenly became non-linear; as these strains were well below the yield point of the material, it can be concluded that Cone 1 failed elastically, i.e. its non-linearity was geometrical. From Figs 14 and 15 it can be seen that for both Cones 2 and 3, especially for the latter, the recorded strains were near the yield point, thereby indicating possible failure by inelastic instability. In the case of Cone 3, both the m a x i m u m and minimum experimentally recorded strains indicated recorded strains near yield, and Fig. 15 appears to show that the nonlinearitv was more of a material nature than a geometrical one, i.e. Cones 2 and 3 were more likely than Cone I to suffer elastic knockdown due to initial imperfections. This was the reason why Cone 3 had an experimentally recorded buckling pressure only slightly higher than that of Cones 1 and 2, despite the fact that it had more stiffening rings and also that these stiffening rings were of greater depth.
Buckling and vibration oJ'ring-stiffened cones under uniform external pressure
333
Buckling presSure
3.0
/ Gauge
4
1
/
2.5tl
2.0v
u~
1.5-
¢,L
~ 1"0X
0.5"
0
-100
-200
-300
-400
-500
-600
-7o0
-sbo
-gbo
-lo~
Strain x 10-6
Fig. 13. Variation of strain with pressure for Cone 1.
Gauge 6 Gauge 10
Buckling pressure
~4II a.
:E ,~3.
,= °" 2
x LU
-160 -2oo -3oo -4bo
-~0
-6bo
-76o -~oo -9bo -~obo -.bo-,206
Strain x 10"6
Fig. 14. Variation of strain with pressure for Cone 2.
334
C. T. F. Ross, T. Jotms
5" --
--
Gauge 6 G a u g e 10 Buckling p r e s s u r e
j l
/J
re n
==
tl
X hi
-200
-400
i
-eoo
-~:>o
-1~o
-l~oo
-1,;oo
S t r a i n x 10 . 6
Fig. 15. Variation of strain with pressure for Cone 3.
: \
I 00/\
Fig. 16. Post-buckled out-of-circularity plot for Cone 1.
Buckling and vibration o]'ring-stif/'ened cones under uni]bnn external pressure
J
"--9
Fig. 17. Post-buckled out-of-circularity plot fl~r Cone 2.
~Z
Fig. 18. Post-buckled out-of-circularity plot for Cone 3.
335
C. T. F. Ross, T. Johns
336
900"
850-
to
X
750.
.c
N 7oo. 650. L,. U
i
!
i
i
Gauge No.
Fig. 19. Variation of hoop strain around a circumference for Cone 1 at a pressure of 2.96 MPa.
1100 ]
/ g
_ 950m
Gauge No.
Fig. 20. Variation of hoop strain around a circumference for Cone 2 at a pressure of 3-875 MPa. Figures 19-21 show plots of the experimental hoop strains around a c o m p l e t e circumference, just prior to buckling, for all three cones, where it can be seen that Cones 1 and 2 buckled with 4 circumferential lobes and C o n e 3 buckled with either 3 or 4 circumferential lobes. These figures also a p p e a r to lend weight to the heuristic arguments of the previous paragraph. F o r example, Fig. 19 shows that Cone 1 had the purest pre-buckling
Buckling and vibration of ring-stiffened cones under uniform external pressure
337
1500"
1450"
/
1400,
1350
x 1300,
X
c
1250.
1200.
1150
;
;
;
g
Gauge No.
g
7
s
;
1o
Fig. 21. V a r i a t i o n o f h o o p strain a r o u n d a circumference for C o n e 3 at a pressure of 4.05 MPa.
circumferential mode, thereby indicating that it failed through elastic instability. Furthermore, from the experimental buckling modes for Cones 2 and 3, it can be seen that the pre-buckling mode of Cone 3 was considerably less pure than that of Cone 2, and that for Cone 3 material non-linearity played an important role. The experimentally obtained buckling pressure for Cone 1 was 2.98 MPa, whilst those for Cones 2 and 3 were 3.93 MPa and 4.1 MPa, respectively.
C O M P A R I S O N BETWEEN EXPERIMENT AND T H E O R Y Vibration For vibration in air, 14 equal length elements were chosen for Cones I and 2, and 16 equal length elements for Cone 3. The chosen meshes for vibration in water are shown in Fig. 22, where it can be seen that at the fluid-structure interface one fluid element was used for two shell elements, the numbers of shell elements being the same as those used for vibration in air. Figures 23-25 show comparisons between experiment and theory, where
338
C. T. F. Ross, T. Johns
••(a)
F (c)
(b)
Fig. 22. Meshes adopted for fluid/structure. (a) Mesh for Cones 1 and 2 (water external). (b) Mesh for Cone 3 (water external). (c) Mesh for Cones I and 2 (free-flood).
Experiment Theory
3000-
/
x
/
X 2 500"
Free flood
~I / .
=.F. =O00" eO
./
g ~soo-
/~
/ 1000-
500-
n
Fig. 23. Variation of frequency with n (Cone 1).
0
339
Experiment Theory In air Water external Free flood
4000X •
/ //
~ 3000-
/
tJ t0
g /g
//
O" e f,,.
u. 2 000-
1000-
i
I
1
2
|
I
I
3
4
5
n
Fig. 24. Variation of frequency with n (Cone 2).
4 000"
Expert ment Theory In air Water external Free flood
×
//~x //~/ /y
N
C
=0" 2 0 0 0 U.
1 (X)O"
!
|
i
n
Fig. 25. Variation of frequency with n (Cone 3).
340
C. T. F. Ross, T. Johns 3000"
0 2500"
~.
X
Model No.1 Model No.2 Model No.3
x / i ~
2000-
U ¢,.
= 1500o" e b. (.
500.
n
Fig. 26. Variation of frequencywith n (experimentalonly; water internal and air external).
the magnitudes of the resonant frequencies are plotted against n, the n u m b e r of circumferential waves. For underwater vibrations, it can be seen that the magnitudes of the resonant frequencies are considerably less than those obtained in air, and also that the magnitudes of the resonant frequencies in the free flood condition are only fractionally less than for the water external/air internal condition. Figure 26 shows plots of the magnitudes of the experimentally obtained resonant frequencies against n, the number of circumferential waves, for the water internal/air external condition. (The experimental results in Fig. 26 are not of interest in the present study; they are presented as being of possible interest to some readers who may be concerned with this case. ) Static buckling Table 2 shows the comparison between the experimentally observed and theoretically derived buckling pressures based on elastic theory. It can be seen that agreement between experiment and theory is reasonable for Cone 1 but that the theoretical predictions, which are based on perfect vessels
Buckling and vibration of ring-stiffened cones under uniform external pressure
341
TABLE 2 Buckling Pressures for the Three Cones (MPa) Cone
Experiment
Elastic theory
l
2.98 (4)
3.55 (4)
2 3
3.93 (4) 4.10 (3, 4)
5.48 (4) 6.65 (3)
The figures in parentheses represent n.
buckling elastically, grossly overestimate the buckling pressures for Cones 2 and 3. In the case of Cone 2, due to the effects of initial out-of-circularity and the fact that the vessel buckled inelastically, the elastic knockdown was about 39%. In the case of Cone 3, as the vessel was relatively 'thicker' than Cones 1 and 2, the elastic knockdown was larger, about 62.3%. Another factor which caused the theoretical buckling pressures to be higher than those obtained experimentally was that there may have been some elastic relaxation, in the axial direction, at the ends of the vessels. For example, the theory assumed that the vessels were completely fixed at their large ends and clamped at their small ends but, as each end was secured by only four bolts, there may have been some elastic relaxation, causing the experimentally obtained buckling pressures to be less than if more bolts had been used. This effect, however, was likely to be small, as the experimentally obtained buckling pressure for Cone 1, which buckled elastically, was only about 19% less than elastic theory predictions. Higher buckling pressures for Cones 2 and 3 could have been achieved if these vessels had been constructed with a higher degree of precision from a higher tensile material.
CONCLUSIONS The vibration investigation has shown that the effects of water drastically reduced the magnitudes of the resonant frequencies that were obtained in air. The results have also shown that, if a vessel suffers inelastic instability, its experimentally obtained buckling pressures can be considerably less than those predicted by elastic theory. In fact, when vessels suffer inelastic instability, increasing the size of their stiffening rings need not increase their buckling resistances appreciably. The lower parts of Figs 8-10 show that the position of the buckled lobe did not necessarily occur at the position of maximum inward out-of-circularity.
342
C. T. F. Ross, T. Johns REFERENCES
1. Ross, C. T. F. and Johns, T., Vibration of hemi-ellipsoidal axisymmetric domes submerged in water, Proc. Inst. Mech. Engrs, 200 (1986) 389--98. 2. Ross, C. T. F., Johns, T. and Emile, Dvnamic buckling of thin-walled domes under external water pressure, (2onference on Applied Solid Mechanics--2, University of Strathclyde, April 1987. 3. Port, K. F. and Ross, C. T. F., Free vibration of submerged thin-walled domes. Conference on Recent Advances in Structural Dynamics, University of Southampton, 1980. 4. Ross, C. T. F. and Johns, T., Vibration of submerged hemi-ellipsoidal domes, J. Sound Vibr., 91 (1983) 363-73. 5. Ross, C. T. F., Emile and Johns, T., Vibration of thin-walled domes under external water pressure, J. Sound Vibr., 114 (1987) 453--63. 6. Ross, C. T. F., Lobar buckling of thin-walled cylindrical and truncated conical shells under external pressure, S N A M E J. Ship Research, 18 (1974) 272-7. 7. Ross, C. T. F., Finite elements for the vibration of cones and cylinders, IJNME, 9 (1975) 833-45. 8. Ross, C. T. F., Vibration and instability of ring-reinforced circular cylindrical and conical shells, SNAMEJ. Ship Research, 20 (1976) 22-31. 9. Ross, C. T. F., Finite Element Programs for Axisymmetric Problems in Engineering, Horwood, Chichester, 1984. 10. Zienkiewicz, O. C. and Newton, R. E., Coupled vibrations of a structure submerged in a compressible fluid, International Symposium on Finite Element Techniques, University of Stuttgart, 1969.
Buckling and Vibration of Ring-Stiffened Cones under Uniform External Pressure C. T. F. Ross & T. J o h n s Department of Mechanical Engineering, Portsmouth Polytechnic, Anglesca Building, Anglesea Road, Portsmouth POI 3D J, UK (Received 10 July 1987; accepted 4 December 1987)
ABSTRACT Theoretical and experimental results are presented for the buckling and vibration of three ring-stiffened thin-walled cones constructed from mild steel. For vibration, the study consisted of exciting the cones in air and while partially and fully submerged in water. For the buckling study, the cones were subjected to a gradually applied uniform external pressure until instability' occurred.
INTRODUCTION The vibration of ring-stiffened cones under water is of much importance in the design of submarine pressure hulls, off-shore drilling rigs and other similar structures. Such vibrations can be caused by a number of different periodic forces, including those due to the motion of the surrounding fluid and those due to out-of-balance machinery. The modal patterns of ringstiffened cones, due to these periodic forces, are numerous but the present study will be concerned only with the lobar modes of vibration of Fig. 1. The reason for this is partly because these modes correspond to the static buckling modes of ring-stiffened cones under uniform external pressure (Fig. 2), and partly because these modes of vibration are the fundamental modes. It has been found in an investigation on the buckling and vibration of domes ~'5 that, as the external hydrostatic pressure is increased for such vessels, their resonant frequencies decrease, and that as the static buckling pressure is approached the fundamental eigenmodes of vibration become 32l
Thin-Walled Structures 0263-8231/88/$03-50 © 1988 Elsevier Science Publishers ltd. England. Printed in Great Britain
322
L'. 1". F. R,.,~. |
F. J~.,hpt.~ I
l
Fig. 1. Lobar eigenmode of vibration.
Fig. 2. General instability of a ring-stiffened cone.
very similar to the static buckling eigenmodes. Thus it is possible that, when the vibration eigenmodes are of similar form to the static buckling eigenmodes, any small exciting force may trigger off dynamic buckling at a pressure which may be considerably less than the static buckling pressure. Under uniform external pressure, ring-stiffened cones can fail through general instability, where the ring-shell combination buckles bodily, as shown in Fig. 2. The strength of these vessels is dependent on the material
BzLcklittg and vibration of ring-stitCh'tied cymes t~tzder tttr(li)rm external presst~re
323
and geometrical properties of the cone, and two major factors that influence the strength of these vessels are the size and the number of stiffening rings. Initial out-of-circularity of these vessels can also affect their buckling resistances, although this effect is, in general, less serious for long thin shells made from high tensile materials than for shorter and thicker shells made from materials of lower tensile strength. In the present paper, initial out-ofcircularity is defined as the radial departure (plus and minus) from the mean circle. The buckling pressures of near perfect long thin shells constructed from high tensile materials can very often be predicted precisely from elastic instability considerations. For vessels that buckle inelastically, both out-ofcircularity and plastic behaviour of the material of construction must be taken into consideration to determine the magnitude of the elastic knockdown. Elastic knockdown can be described as the difference between the experimentally obtained buckling pressure and the theoretical buckling pressure predicted for a perfect vessel failing through elastic instability. In the near future it is hoped to extend the present investigation to study dynamic buckling, where the modes of vibration can interact with the static buckling mode to cause the vessel to fail at a pressure appreciably less than
Fig. 3. Ring-stiffened cones.
324
{. i /'. h:n~s, l. Jol';:',
that required to cause static buckling. Such an investigation has tflrcad', been successfully carried out on a family of hemi-ellipsoidal domes. ' " The experimental and theoretical investigations of the present study wcrc based on the three machine stiffened thin-~valled cones of Fig. 3, and involved the following: (a) Vibration of the cones in air. (b) Vibration of the cones with water on their external surfaces and air on their internal surfaces. (c) Vibration of the cones fully submerged in water (free-flooded). (d) Vibration of the cones with water on their internal surfaces and air on their external surfaces. (e) Static buckling of the ring-stiffened cones under uniform external pressure.
T H E O R E T I C A L ANALYSIS The truncated conical shell element that was used for this analysis has been described elsewhere, ~ and the ring element was also the same as that previously described? The conical shell element was defined by ring nodes at its ends. as shown in Fig. 4, and it had four degrees of freedom per node, namely, u~, v, w,~and 0,. The ring element had three degrees of freedom, namely, v, ~t~ and O~,and it is shown in Fig. 5, where e is the distance of the centroid of the ringstiffener from the mid-surface of the shell (negative if the frame is external). For both the conical shell element and the ring stiffener element, the w
0
A
~=1 V
ij
Ri
Rj
r
I ~..Axis oflsymmetry oo
t
P a r t section on
A--A
A Fig. 4. Conical element.
Buckling and vibration of ring-stiffened cones under uniform external pressure
I
325
I
Fig. 5. F r a m e with shell.
5
/
tJ
---
6
7q
r
|
2
Fig. 6. C r o s s - s e c t i o n o f a n n u l a r fluid e l e m e n t .
stiffness and mass matrices were based on small deflection elastic theory, but the geometrical stiffness matrices were based on large deflection elastic theory. In o r d e r to represent the motion of the fluid, a 'solid' annular element was used, as shown in Fig. 6. The cross-section of this element was an 8-node isoparametric type, and each node had one pressure degree of freedom. The e l e m e n t is described in greater detail in Refs 1-5. The interaction at the fluid-structure interface was c a r d e d out by coupling the solid and fluid elements at their c o m m o n boundary in the manner described by Zienkiewicz and Newton. t0
EXPERIMENTAL METHOD T h e vessels were machined from a solid block of mild steel, EN1A. Their internal surfaces were machined first, and then each vessel was snugly fitted on to a conical mandrel where its external surface was machined to the nominal g e o m e t r y given in Table 1 and Fig. 7, where N = number of ring-stiffeners b = width of ring-stiffener, of 'rectangular' cross-section
326
c . T. F. Ross, T. Johns
d e R~ R,. h n
= = = = = =
d e p t h o f ring-stiffener, o f ' r e c t a n g u l a r " cross-section ring e c c e n t r i c i t y (negative as the rings were external) r a d i u s o f shell at small e n d = 1.905 cm r a d i u s o f shell at large e n d = 5.08 cm shell t h i c k n e s s = 0.635 m m n u m b e r o f c i r c u m f e r e n t i a l waves o r lobes
The manufacture
of these vessels was extremely good, and plots of the initial
out-of-circularity before and after the vibration tests were carried out are s h o w n in F i g s 8 - 1 0 .
TABLE 1
Geometrical Properties of the Cones M o d e l no.
N
b (ram)
d (ram)
e (ram)
I 2 3
6 6 7
1.016 1.016 1.016
i-016 1.524 2-032
-0.826 - 1.080 - 1.333
t
211.0
r---
)
o
1
I
J
f
J
(:>635 Cone no.
A
B
I 2
I-0 1.0
1.0 1-5
:z
1.o
2.0
Fig. 7. Dimensions (nominal) of the ring-stiffened cones (mm).
Buckling and vibration of ring-stiffened cones under uniform external pressure
(a)
{b)
Befor~ Vibration
After Vibration
Fig. 8. Initial out-of-circularity plots for Cone I.
327
328
C. T. F. Ross, T. Johns
(a)
Befo~ Vibration
( {b)
After Vibration
Fig. 9. Initial out-of-circularity plots for Cone 2.
Buckling and vibration of ring-stiffened cones under uniform external pressure
(a)
(b)
Before Vibration
After Vibration
Fig. 10. Initial out-of-circularity plots for Cone 3.
329
330
C. T. F. Ross, 7". Johns
In Figs 8 and 9 the out-of-circularity plots were taken midway between the third and fourth rings, on the external surfaces of the vessels. In Fig. l0 the out-of-circularity plots were measured at a distance of 11-75 cm from the larger end of the vessel, on its external surface. The importance of taking these out-of-circularity plots was to investigate whether or not the maximum buckled form occurred in the vicinity of the largest initial imperfection. F r o m the plots it can be seen that the vibration tests had little or no effect on improving or worsening the out-of-circularity. It can also be seen that the worst out-of-circularity was that of Cone 1, =0.0127 mm. The initial out-ofcircularity for Cone 2 was =2-5 x 10 -3 mm and that for Cone 3 was =6.4 × 10 -3 mm. The elastic (Young's) modulus (E) was determined experimentally as 1.9 x 10 t' N m -2, and the density (p) and Poisson's ratio (u) were assumed to be 7800 kg m -3 and 0.3 respectively.
VIBRATION Prior to vibrating the cones, initial out-of-circularity plots were taken of all three cones as shown in the upper parts of Figs 8-10. T h e cones were excited with two electromagnetic shakers and with the aid of a frequency response analyser and other associated electronic equipment, as previously described. ,-5 Even lobar modes were obtained by keeping the shakers in-phase, and odd lobar modes were obtained by keeping the shakers 180° out-of-phase. W h e n the cones were vibrated in air, a microp h o n e was used; underwater the microphone was replaced by a hydrophone. For the underwater tests, except for the case of water on the internal surface only, the experiments were carried out in a tank of diameter 1.5 m, as shown in Fig. 11. A f t e r vibrating the cones and before the static buckling tests were c o m m e n c e d , 'initial' out-of-circularity plots were taken again, and these showed that the vibration tests had not affected the initial out-of-circularity appreciably, as shown in the lower parts of Figs 8-10. These figures also show the positions where buckling occurred when the vessels were tested to destruction.
BUCKLING
T h e large diameter end of each cone was bolted to the top of the tank with the aid o f four bolts, and the smaller end of each cone was blocked off by a closure plate, with the aid of another four bolts, as shown in Fig. 12, so it can
Buckling and ~'ibratiott o f ring-stit?ened cr_mes trader tmt torm exterpTal pre'ss"~re
331
Fig. 11. L;ndcr~atcr vibrtltion tests.
be concluded that the larger diameter end was effectively fixed but that the smaller diameter end was "clamped'. T h e r e were four degrees of freedom at each node. namely, an ~xi,l deflection (u°), a radial deflection (w°), a circumferential deflection (~) and a rotational displacement (0), as shown in Fig. 4. In the "clamped" condition. the radial, circumferential and rotational displacements are zero. but the axial displacement is not. To assist in determining the circumferential eigcnmodes due to buckling. each cone had attached to its internal surface ten electrical resistance strain gauges, placed in a circumferential direction, in the region where it xvas expected that the buckled mode would have its largest deflection. Plots of pressure against strain for the strain gauges recording the largest and smallest magnitudes of strain, just prior to buckling, are shown in Figs 13-15. Figures 16-18 show post-buckled forms of the vessels, taken at the positions of the maximum deflection, together with the locations of the strain gauges. These out-of-circularity plots were taken on the outer surfaces of the vessels, where it can be clearly seen that Cone 1 buckled ~vith four circumferential lobes. The out-of-circularity plots for Cones 2 and 3 do not reveal the n u m b e r of circumferential lobes that these two vessels buckled into. F r o m Fig. 13 it can be seen that. just prior to buckling, the measured
332
C I" I-" R,_),~. I J~,~i~:.,
Air w n t
Top plate
Ring
'0' Ring
Ring stiffened cone W~ter
vessel 'O'
Ring
:1 blanking plug
Fig. 12. C o n e in tc~,t tank.
strains suddenly became non-linear; as these strains were well below the yield point of the material, it can be concluded that Cone 1 failed elastically, i.e. its non-linearity was geometrical. From Figs 14 and 15 it can be seen that for both Cones 2 and 3, especially for the latter, the recorded strains were near the yield point, thereby indicating possible failure by inelastic instability. In the case of Cone 3, both the m a x i m u m and minimum experimentally recorded strains indicated recorded strains near yield, and Fig. 15 appears to show that the nonlinearitv was more of a material nature than a geometrical one, i.e. Cones 2 and 3 were more likely than Cone I to suffer elastic knockdown due to initial imperfections. This was the reason why Cone 3 had an experimentally recorded buckling pressure only slightly higher than that of Cones 1 and 2, despite the fact that it had more stiffening rings and also that these stiffening rings were of greater depth.
Buckling and vibration oJ'ring-stiffened cones under uniform external pressure
333
Buckling presSure
3.0
/ Gauge
4
1
/
2.5tl
2.0v
u~
1.5-
¢,L
~ 1"0X
0.5"
0
-100
-200
-300
-400
-500
-600
-7o0
-sbo
-gbo
-lo~
Strain x 10-6
Fig. 13. Variation of strain with pressure for Cone 1.
Gauge 6 Gauge 10
Buckling pressure
~4II a.
:E ,~3.
,= °" 2
x LU
-160 -2oo -3oo -4bo
-~0
-6bo
-76o -~oo -9bo -~obo -.bo-,206
Strain x 10"6
Fig. 14. Variation of strain with pressure for Cone 2.
334
C. T. F. Ross, T. Jotms
5" --
--
Gauge 6 G a u g e 10 Buckling p r e s s u r e
j l
/J
re n
==
tl
X hi
-200
-400
i
-eoo
-~:>o
-1~o
-l~oo
-1,;oo
S t r a i n x 10 . 6
Fig. 15. Variation of strain with pressure for Cone 3.
: \
I 00/\
Fig. 16. Post-buckled out-of-circularity plot for Cone 1.
Buckling and vibration o]'ring-stif/'ened cones under uni]bnn external pressure
J
"--9
Fig. 17. Post-buckled out-of-circularity plot fl~r Cone 2.
~Z
Fig. 18. Post-buckled out-of-circularity plot for Cone 3.
335
C. T. F. Ross, T. Johns
336
900"
850-
to
X
750.
.c
N 7oo. 650. L,. U
i
!
i
i
Gauge No.
Fig. 19. Variation of hoop strain around a circumference for Cone 1 at a pressure of 2.96 MPa.
1100 ]
/ g
_ 950m
Gauge No.
Fig. 20. Variation of hoop strain around a circumference for Cone 2 at a pressure of 3-875 MPa. Figures 19-21 show plots of the experimental hoop strains around a c o m p l e t e circumference, just prior to buckling, for all three cones, where it can be seen that Cones 1 and 2 buckled with 4 circumferential lobes and C o n e 3 buckled with either 3 or 4 circumferential lobes. These figures also a p p e a r to lend weight to the heuristic arguments of the previous paragraph. F o r example, Fig. 19 shows that Cone 1 had the purest pre-buckling
Buckling and vibration of ring-stiffened cones under uniform external pressure
337
1500"
1450"
/
1400,
1350
x 1300,
X
c
1250.
1200.
1150
;
;
;
g
Gauge No.
g
7
s
;
1o
Fig. 21. V a r i a t i o n o f h o o p strain a r o u n d a circumference for C o n e 3 at a pressure of 4.05 MPa.
circumferential mode, thereby indicating that it failed through elastic instability. Furthermore, from the experimental buckling modes for Cones 2 and 3, it can be seen that the pre-buckling mode of Cone 3 was considerably less pure than that of Cone 2, and that for Cone 3 material non-linearity played an important role. The experimentally obtained buckling pressure for Cone 1 was 2.98 MPa, whilst those for Cones 2 and 3 were 3.93 MPa and 4.1 MPa, respectively.
C O M P A R I S O N BETWEEN EXPERIMENT AND T H E O R Y Vibration For vibration in air, 14 equal length elements were chosen for Cones I and 2, and 16 equal length elements for Cone 3. The chosen meshes for vibration in water are shown in Fig. 22, where it can be seen that at the fluid-structure interface one fluid element was used for two shell elements, the numbers of shell elements being the same as those used for vibration in air. Figures 23-25 show comparisons between experiment and theory, where
338
C. T. F. Ross, T. Johns
••(a)
F (c)
(b)
Fig. 22. Meshes adopted for fluid/structure. (a) Mesh for Cones 1 and 2 (water external). (b) Mesh for Cone 3 (water external). (c) Mesh for Cones I and 2 (free-flood).
Experiment Theory
3000-
/
x
/
X 2 500"
Free flood
~I / .
=.F. =O00" eO
./
g ~soo-
/~
/ 1000-
500-
n
Fig. 23. Variation of frequency with n (Cone 1).
0
339
Experiment Theory In air Water external Free flood
4000X •
/ //
~ 3000-
/
tJ t0
g /g
//
O" e f,,.
u. 2 000-
1000-
i
I
1
2
|
I
I
3
4
5
n
Fig. 24. Variation of frequency with n (Cone 2).
4 000"
Expert ment Theory In air Water external Free flood
×
//~x //~/ /y
N
C
=0" 2 0 0 0 U.
1 (X)O"
!
|
i
n
Fig. 25. Variation of frequency with n (Cone 3).
340
C. T. F. Ross, T. Johns 3000"
0 2500"
~.
X
Model No.1 Model No.2 Model No.3
x / i ~
2000-
U ¢,.
= 1500o" e b. (.
500.
n
Fig. 26. Variation of frequencywith n (experimentalonly; water internal and air external).
the magnitudes of the resonant frequencies are plotted against n, the n u m b e r of circumferential waves. For underwater vibrations, it can be seen that the magnitudes of the resonant frequencies are considerably less than those obtained in air, and also that the magnitudes of the resonant frequencies in the free flood condition are only fractionally less than for the water external/air internal condition. Figure 26 shows plots of the magnitudes of the experimentally obtained resonant frequencies against n, the number of circumferential waves, for the water internal/air external condition. (The experimental results in Fig. 26 are not of interest in the present study; they are presented as being of possible interest to some readers who may be concerned with this case. ) Static buckling Table 2 shows the comparison between the experimentally observed and theoretically derived buckling pressures based on elastic theory. It can be seen that agreement between experiment and theory is reasonable for Cone 1 but that the theoretical predictions, which are based on perfect vessels
Buckling and vibration of ring-stiffened cones under uniform external pressure
341
TABLE 2 Buckling Pressures for the Three Cones (MPa) Cone
Experiment
Elastic theory
l
2.98 (4)
3.55 (4)
2 3
3.93 (4) 4.10 (3, 4)
5.48 (4) 6.65 (3)
The figures in parentheses represent n.
buckling elastically, grossly overestimate the buckling pressures for Cones 2 and 3. In the case of Cone 2, due to the effects of initial out-of-circularity and the fact that the vessel buckled inelastically, the elastic knockdown was about 39%. In the case of Cone 3, as the vessel was relatively 'thicker' than Cones 1 and 2, the elastic knockdown was larger, about 62.3%. Another factor which caused the theoretical buckling pressures to be higher than those obtained experimentally was that there may have been some elastic relaxation, in the axial direction, at the ends of the vessels. For example, the theory assumed that the vessels were completely fixed at their large ends and clamped at their small ends but, as each end was secured by only four bolts, there may have been some elastic relaxation, causing the experimentally obtained buckling pressures to be less than if more bolts had been used. This effect, however, was likely to be small, as the experimentally obtained buckling pressure for Cone 1, which buckled elastically, was only about 19% less than elastic theory predictions. Higher buckling pressures for Cones 2 and 3 could have been achieved if these vessels had been constructed with a higher degree of precision from a higher tensile material.
CONCLUSIONS The vibration investigation has shown that the effects of water drastically reduced the magnitudes of the resonant frequencies that were obtained in air. The results have also shown that, if a vessel suffers inelastic instability, its experimentally obtained buckling pressures can be considerably less than those predicted by elastic theory. In fact, when vessels suffer inelastic instability, increasing the size of their stiffening rings need not increase their buckling resistances appreciably. The lower parts of Figs 8-10 show that the position of the buckled lobe did not necessarily occur at the position of maximum inward out-of-circularity.
342
C. T. F. Ross, T. Johns REFERENCES
1. Ross, C. T. F. and Johns, T., Vibration of hemi-ellipsoidal axisymmetric domes submerged in water, Proc. Inst. Mech. Engrs, 200 (1986) 389--98. 2. Ross, C. T. F., Johns, T. and Emile, Dvnamic buckling of thin-walled domes under external water pressure, (2onference on Applied Solid Mechanics--2, University of Strathclyde, April 1987. 3. Port, K. F. and Ross, C. T. F., Free vibration of submerged thin-walled domes. Conference on Recent Advances in Structural Dynamics, University of Southampton, 1980. 4. Ross, C. T. F. and Johns, T., Vibration of submerged hemi-ellipsoidal domes, J. Sound Vibr., 91 (1983) 363-73. 5. Ross, C. T. F., Emile and Johns, T., Vibration of thin-walled domes under external water pressure, J. Sound Vibr., 114 (1987) 453--63. 6. Ross, C. T. F., Lobar buckling of thin-walled cylindrical and truncated conical shells under external pressure, S N A M E J. Ship Research, 18 (1974) 272-7. 7. Ross, C. T. F., Finite elements for the vibration of cones and cylinders, IJNME, 9 (1975) 833-45. 8. Ross, C. T. F., Vibration and instability of ring-reinforced circular cylindrical and conical shells, SNAMEJ. Ship Research, 20 (1976) 22-31. 9. Ross, C. T. F., Finite Element Programs for Axisymmetric Problems in Engineering, Horwood, Chichester, 1984. 10. Zienkiewicz, O. C. and Newton, R. E., Coupled vibrations of a structure submerged in a compressible fluid, International Symposium on Finite Element Techniques, University of Stuttgart, 1969.