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1. 6-1
Buckling of Shells—Pitfall for
Designers
David Bushnell
AIAA Journal, Vol. 19, No. 9, 1183–1226

2. 6-2
Roadmap of This Talk
1. A rogue’s gallery of examples
2. Classical buckling and imperfection sensitivity
3. Nonlinear collapse and linear bifurcation model
4. Bifurcation buckling from nonlinear
prebuckling state
5. Effect of boundary conditions
6. Examples of stable post-buckling
7. Interaction of local and general instability
8. Effect of imperfections on stiffened shells

3. 6-3
Buckling Is Indeed a Mysterious,
Awe-Inspiring Phenomenon!

4. 6-4
Some Structures May Be Grossly Underdesigned
If Buckling Is Not Properly Accounted For

5. 6-5
Building Collapse

6. 6-6
Collapse of Vertical Tank
Collapse
due to
unexpected
buckling of
lower head
under
internal
pressure

7. 6-7
How on Earth Did This Happen?

8. 6-8
Water Tank in Belgium Collapsed

9. 6-9
Schematic of Belgian Water Tank

10. 6-10
Collapse Initiated Near Bottom of Conical
Water Tank

11. 6-11
Mylar Models of Conical Water Tank
Tested in Belgium

12. 6-12
Schematic of Water Tank & Buckling Mode
Predicted by BOSOR5

13. 6-13
Stainless Steel Wine Tanks Buckled Due
to Earthquake

14. 6-14
More Buckled Wine Tanks

15. 6-15
Yet More Buckled Wine Tanks

16. 6-16
Still More Buckled Wine Tanks

17. 6-17
Large Water Tank Buckled During San
Fernando Earthquake, 1971
Axisymmetric
elephant foot
buckling

18. 6-18
Another Buckled Water Tank
Buckling occurs where wall thickness
is stepped down from course below

19. 6-19
Another View of Buckling of Same Large
Water Tank

20. 6-20
Prebuckling Deformation of Nuclear Containment
Shell Due to Horizontal Ground Motion
Buckling will
occur here
due to
maximum
axial
compression

21. 6-21
Buckling of Nuclear Containment Shell
Predicted by BOSOR4
Non-
axisymmetric
buckles

22. 6-22
Nuclear Reactor Cooling Towers
Nuclear reactor cooling towers are large
shells that can buckle from wind loading

23. 6-23
Cooling Towers Are Huge Shells!

24. 6-24
Laying Oil Pipeline at Sea
Buckling can occur on
compressive side of bent
pipe on bottom of pipe;
here, top of pipe where it
bends other way near
sea bed

25. 6-25
Offshore Oil Platform Support
Supporting
truss consists
of assemblage
of thin
cylindrical
shells with
large diameters

26. 6-26
One of Large Joints of Supporting Truss
of Oil Platform

27. 6-27
Offshore Oil Platform Battered by Huge Wave
Wave action
can cause
buckling of
supporting
members of
oil platform

28. 6-28
Liquid Natural Gas (LNG) Vessel
5 large spherical LNG tanks

29. 6-29
Large LNG Tank
Spherical
tank will be
supported at
its equator
when
installed in
LNG tanker

30. 6-30
Possible Buckling of Partially Filled LNG Tank
Buckling is
from hoop
compression
just below
the equator
Prebuckling
stress state is
meridional
tension
combined
with hoop
compression

31. 6-31
Thin Shell Space Structures
Thin shell space structures for use in orbit
Many of these lightweight structures are designed by buckling

32. 6-32
Space shuttle external tank
Aerospace
structures like
this may buckle
due to launch
loads combined
with
circumferentially
varying dynamic
pressure

33. 6-33
Large Stiffened Cylindrical Shell

34. 6-34
Payload Shroud Consists of Several
Segments with Joints
The thicknesses
of the shell walls
varies along the
length because
the destabilizing
stresses
generated by the
launch loads vary
along the length
of the structure

35. 6-35
Launch Loads & Buckling Mode

36. 6-36
Schematic of Local Buckling Near
Joint in Payload Shroud
Centerline of
cylindrical
shell is on
right-hand
side
Loading is
axial
compression

37. 6-37
Local Buckles in Joint Region

38. 6-38
Brain with “Buckled” Folds

39. 6-39
Another View of “Buckled” Folds in Human Brain

40. 6-40
Falstaff’s Boots Have Buckled!

41. 6-41
Plastic Buckling of Axially Compressed Thick
Cylindrical Shell

42. 6-42
Buckling of Perfect & Imperfect Shell
In contrast to
the behavior
shown in the
previous slide,
here bifurcation
buckling, B,
occurs before
axisymmetric
collapse, A

43. 6-43
Buckling of Perfect & Imperfect Shell With
Greater Amount of Imperfection Sensitivity

44. 6-44
Summary of Analysis Tools
ABAQUS, NASTRAN

45. 6-45
Buckling of Thin Cylindrical Shell Under Uniform
Axial Compression
The buckles are
widespread & small
compared to a
typical structural
dimension. This
behavior indicates
extreme sensitivity
of the critical load
to initial shape
imperfections

46. 6-46
Comparison of Test & Theory for Axially
Compressed Thin Cylindrical Shells
(Perfect shell)
Design
recommendation

47. 6-47
Buckling of Externally Pressurized Thin
Spherical Shell
Mandrel inside
shell prevents
collapse
Critical loads of
shells with this
type of buckling
are extremely
sensitive to initial
shape
imperfections

48. 6-48
Buckling of Externally Pressurized
Spherical Caps
Deeper caps
behave more
like complete
spherical
shells
Their critical
pressures are
therefore more
sensitive to
initial shape
imperfections
than are those
for shallow
caps

49. 6-49
Comparison of Test & Theory for Buckling of
Spherical Caps Under Uniform External Pressure

50. 6-50
Asymptotic Imperfection Sensitivity Factor, b,
from Koiter Theory

51. 6-51
Axisymmetric Elastic-plastic Buckling of Axially
Compressed Thick Cylindrical Shell

52. 6-52
Buckling of Unpressurized & Pressurized Thin
Cylindrical Shells Under Torque

53. 6-53
Buckling of Thin Cylindrical Shells
Under Torque
Imperfection
sensitivity is
much milder
under torque
than under
axial
compression

54. 6-54
Buckling of Externally Pressurized Ring-
Stiffened Cylindrical Shells
Strong rings
Medium rings Weak rings

55. 6-55
Externally Pressurized Cylindrical Shells
Comparison of
test & theory
Koiter
imperfection
sensitivity
parameter, b

56. 6-56
Buckling of Axially Compressed Stiffened
Cylindrical Shell
Tested by Professor Josef
Singer, et al (Technion)
Critical loads of stiffened
shells are less sensitive to
initial imperfections than
monocoque shells
because typical buckle
size is comparable to size
of test specimen: effective
thickness for axial bending
is large

57. 6-57
Buckling of Axially Stiffened Cylindrical
Shells Under Axial Compression
Buckling of
perfect
cylindrical
shells with
outside vs.
inside
stringers
Koiter
imperfection
sensitivity
parameter, b

58. 6-58
Buckling of Perfect & Imperfect 3-Layered
Composite Axially Compressed Cylindrical Shells
Note that critical
load of strongest
shell (40°) is
most sensitive
to amplitude of
initial
imperfection

59. 6-59
Bifurcation Buckling Model v. Nonlinear
Collapse Model
Normal
load, P (lb)

60. 6-60
Another Bifurcation Model v. Collapse

61. 6-61
Axially Compressed Cylindrical Shell With
Rectangular Cutout
Note initial buckling
near vertical edge of
cutout
Buckling on either
side of cutout causes
shedding of axial
load to rest of shell,
which continues to
accept more axial
load
Test by Almroth

62. 6-62
State of Shell at Collapse
Shell
continues to
accept more
& more axial
load until
cylindrical
wall buckles
everywhere

63. 6-63
Yet Another Example of Bifurcation
Buckling Model v. Collapse

64. 6-64
Buckling of Torispherical Shell Under Internal
Pressure
Under internal pressure, knuckle
region is under meridional
tension combined with hoop
compression
Therefore, buckles are elongated
in meridional direction
First buckle forms, relieving
compressive hoop stress in its
neighborhood, permitting further
loading of shell as additional
isolated buckles form one by one
as internal pressure is further
increased

65. 6-65
Prebuckled Hoop Stress
Prebuckled
hoop stress in
internally
pressurized
torispherical
shell from
linear v.
nonlinear
theory

66. 6-66
Buckled Aluminum & Mild Steel Internally
Pressurized Torispherical Heads (Galletly)
Aluminum head
Mild steel head

67. 6-67
Buckled Externally Pressurized Cylinder Shells
Rolled & welded Machined

68. 6-68
Simulation of Fabrication Process
Thru-thickness stress distribution after
rolling the skin to radius smaller than final
radius
Thru-thickness stress distribution after
springback to final radius, Rnominal
Stress distribution after springback & after
welding rings to exterior shell wall
Stress distribution after application of
pressure

69. 6-69
Buckling of Spherical Shells with Edge Ring of
3 Different Sizes

70. 6-70
Collapse of Conical Sandwich Shell
In case of
sandwich wall
construction,
effect of
transverse shear
deformation
(TSD) is very
important

71. 6-71
Development of Shear Buckles & Diagonal
Tension in Web of Beam in Bending
Undeformed beam
Buckled web of bent
beam

72. 6-72
Buckling of Spherical Shell With Concentrated
Load

73. 6-73
Comparison of Test & Theory for Buckling of
Spherical Shell With Inward Concentrated Load

74. 6-74
Glider With Locally Buckled Wing Skin

75. 6-75
Ring-stiffened Shallow Conical Shell
Designed for Entry Into Atmosphere of Mars

76. 6-76
Comparison of Test with 2 Models of
Mars Entry Conical Shell

77. 6-77
Mars Entry Shell—Rings Modeled
As Segmented Shell Branches

78. 6-78
Stiffened Panels for Which Buckling Modal
Interaction May Occur

79. 6-79
Modal Interaction in Buckling of 2-Flanged
Column
Maximum imperfection sensitivity is near the design for
which local & overall buckling occur at the same load.
Flange width, b
(Euler buckling)
(Flange buckling)

80. 6-80
OPTIMIZATION OF AN AXIALLY
COMPRESSED RING AND STRINGER
STIFFENED CYLINDRICAL SHELL
WITH A GENERAL BUCKLING MODAL
IMPERFECTION
AIAA Paper 2007-2216
David Bushnell, Fellow, AIAA, retired

81. 6-81
General buckling mode from STAGS
External T-stringers,
Internal T-rings,
Loading: uniform axial
compression with axial
load, Nx = -3000 lb/in
This is a STAGS model.
50 in.
75 in.

82. 6-82
TWO MAJOR EFFECTS OF A GENERAL
IMPERFECTION
1. The imperfect shell bends when any loads
are applied. This “prebuckling” bending
causes redistribution of stresses between
the panel skin and the various segments of
the stringers and rings.
2. The “effective” radius of curvature of
the imperfect and loaded shell is larger
than the nominal radius: “flat” regions
develop.

83. 6-83
Loaded imperfect cylinder
Maximum
stress,
sbar(max)
=66.87 ksi
“Flat”
region

84. 6-84
The entire deformed cylinder

85. 6-85
The area of maximum stress

86. 6-86
The “flattened” region

87. 6-87
SOME MAIN POINTS
1. Get a “feel” for buckling from many examples.
2. If your structure has a region of compression
buckling is possible. Watch out!
3. There are two kinds of static buckling:
a. nonlinear collapse
b. bifurcation buckling.
4. There are two phases of a buckling problem:
a. prebuckling stress analysis
b. eigenvalue problem.

88. 6-88
MORE MAIN POINTS
5. Imperfections often have a huge influence.
6. Boundary conditions often have a huge influence.
7. A linear bifurcation buckling model is sometimes
a poor predictor of load-carrying capacity.
8. Some structures are stable after buckling first occurs.
9. The load-carrying capacity of optimally designed
structures is often reduced more by imperfections
than is so for other structures.