Thin Shell Buckling Pit-Falls - NASA

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Buckling of Shells—Pitfall for Designers

David Bushnell

AIAA Journal, Vol. 19, No. 9, 1183–1226

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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

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Buckling Is Indeed a Mysterious, Awe-Inspiring Phenomenon!

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Some Structures May Be Grossly Underdesigned If Buckling Is Not Properly Accounted For

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Building Collapse

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Collapse of Vertical Tank

Collapse due to unexpected buckling of lower head under internal pressure

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How on Earth Did This Happen?

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Water Tank in Belgium Collapsed

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Schematic of Belgian Water Tank

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Collapse Initiated Near Bottom of Conical Water Tank

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Mylar Models of Conical Water Tank Tested in Belgium

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Schematic of Water Tank & Buckling Mode Predicted by BOSOR5

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Stainless Steel Wine Tanks Buckled Due to Earthquake

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More Buckled Wine Tanks

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Yet More Buckled Wine Tanks

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Still More Buckled Wine Tanks

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Large Water Tank Buckled During San Fernando Earthquake, 1971

Axisymmetric elephant foot buckling

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Another Buckled Water Tank

Buckling occurs where wall thickness is stepped down from course below

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Another View of Buckling of Same Large Water Tank

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Prebuckling Deformation of Nuclear Containment Shell Due to Horizontal Ground Motion

Buckling will occur here due to maximum axial compression

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Buckling of Nuclear Containment Shell Predicted by BOSOR4

Non-axisymmetric buckles

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Nuclear Reactor Cooling Towers

Nuclear reactor cooling towers are large shells that can buckle from wind loading

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Cooling Towers Are Huge Shells!

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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 nearsea bed

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Offshore Oil Platform Support

Supporting truss consists of assemblage of thin cylindrical shells with large diameters

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One of Large Joints of Supporting Truss of Oil Platform

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Offshore Oil Platform Battered by Huge Wave

Wave action can cause buckling of supporting members of oil platform

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Liquid Natural Gas (LNG) Vessel

5 large spherical LNG tanks

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Large LNG Tank

Spherical tank will be supported at its equator when installed in LNG tanker

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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

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Thin Shell Space Structures

Thin shell space structures for use in orbit

Many of these lightweight structures are designed by buckling

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Space shuttle external tank

Aerospace structures like this may buckle due to launch loads combined with circumferentially varying dynamic pressure

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Large Stiffened Cylindrical Shell

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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

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Launch Loads & Buckling Mode

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Schematic of Local Buckling Near Joint in Payload Shroud

Centerline of cylindrical shell is on right-hand side

Loading is axial compression

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Local Buckles in Joint Region

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Brain with “Buckled” Folds

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Another View of “Buckled” Folds in Human Brain

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Falstaff’s Boots Have Buckled!

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Plastic Buckling of Axially Compressed Thick Cylindrical Shell

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Buckling of Perfect & Imperfect Shell

In contrast to the behavior shown in the previous slide, here bifurcation buckling, B, occurs before axisymmetric collapse, A

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Buckling of Perfect & Imperfect Shell With Greater Amount of Imperfection Sensitivity

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Summary of Analysis Tools

ABAQUS, NASTRAN

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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

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Comparison of Test & Theory for Axially Compressed Thin Cylindrical Shells

(Perfect shell)

Design recommendation

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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

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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

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Comparison of Test & Theory for Buckling of Spherical Caps Under Uniform External Pressure

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Asymptotic Imperfection Sensitivity Factor, b, from Koiter Theory

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Axisymmetric Elastic-plastic Buckling of Axially Compressed Thick Cylindrical Shell

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Buckling of Unpressurized & Pressurized Thin Cylindrical Shells Under Torque

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Buckling of Thin Cylindrical Shells Under Torque

Imperfection sensitivity is much milder under torque than under axial compression

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Buckling of Externally Pressurized Ring-Stiffened Cylindrical Shells

Strong rings

Medium rings Weak rings

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Externally Pressurized Cylindrical Shells

Comparison of test & theory

Koiter imperfection sensitivity parameter, b

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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

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Buckling of Axially Stiffened Cylindrical Shells Under Axial Compression

Buckling of perfect cylindrical shells with outside vs. inside stringers

Koiter imperfection sensitivity parameter, b

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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

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Bifurcation Buckling Model v. Nonlinear Collapse Model

Normal load, P (lb)

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Another Bifurcation Model v. Collapse

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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

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State of Shell at Collapse

Shell continues to accept more & more axial load until cylindrical wall buckles everywhere

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Yet Another Example of Bifurcation Buckling Model v. Collapse

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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

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Prebuckled Hoop Stress

Prebuckled hoop stress in internally pressurized torispherical shell from linear v. nonlinear theory

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Buckled Aluminum & Mild Steel Internally Pressurized Torispherical Heads (Galletly)

Aluminum head

Mild steel head

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Buckled Externally Pressurized Cylinder Shells

Rolled & welded Machined

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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

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Buckling of Spherical Shells with Edge Ring of 3 Different Sizes

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Collapse of Conical Sandwich Shell

In case of sandwich wall construction, effect of transverse shear deformation (TSD) is very important

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Development of Shear Buckles & Diagonal Tension in Web of Beam in Bending

Undeformed beam

Buckled web of bent beam

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Buckling of Spherical Shell With Concentrated Load

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Comparison of Test & Theory for Buckling of Spherical Shell With Inward Concentrated Load

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Glider With Locally Buckled Wing Skin

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

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Comparison of Test with 2 Models of Mars Entry Conical Shell

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Mars Entry Shell—Rings Modeled As Segmented Shell Branches

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Stiffened Panels for Which Buckling Modal Interaction May Occur

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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)

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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

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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.

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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.

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Loaded imperfect cylinder

Maximum stress, sbar(max)=66.87 ksi

“Flat” region

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The entire deformed cylinder

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The area of maximum stress

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The “flattened” region

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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 collapseb. bifurcation buckling.

4. There are two phases of a buckling problem:a. prebuckling stress analysisb. eigenvalue problem.

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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.