Nonlinear Buckling Interaction for Spherical Shells Subject to Pressure and Probing Forces

2017 ◽  
Vol 84 (6) ◽  
Author(s):  
John W. Hutchinson ◽  
J. Michael T. Thompson
Keyword(s):  
Internal Pressure ◽  
Energy Barrier ◽  
Constant Pressure ◽  
External Pressure ◽  
Constant Volume ◽  
Uniform Pressure ◽  
Loading Conditions ◽  
Spherical Shells ◽  
Nonlinear Buckling ◽  
Axisymmetric State

Elastic spherical shells loaded under uniform pressure are subject to equal and opposite compressive probing forces at their poles to trigger and explore buckling. When the shells support external pressure, buckling is usually axisymmetric; the maximum probing force and the energy barrier the probe must overcome are determined. Applications of the probing forces under two different loading conditions, constant pressure or constant volume, are qualitatively different from one another and fully characterized. The effects of probe forces on both perfect shells and shells with axisymmetric dimple imperfections are studied. When the shells are subject to internal pressure, buckling occurs as a nonaxisymmetric bifurcation from the axisymmetric state in the shape of a mode with multiple circumferential waves concentrated in the vicinity of the probe. Exciting new experiments by others are briefly described.

Localization and Propagation of Instabilities in Long Shallow Panels Under External Pressure

1994 ◽  
Vol 61 (4) ◽  
pp. 755-763 ◽  
Author(s):  
T. L. Power ◽  
S. Kyriakides
Keyword(s):  
Constant Pressure ◽  
Limit Load ◽  
Solution Procedure ◽  
External Pressure ◽  
Uniform Pressure ◽  
Membrane Tension ◽  
Pressure Loading ◽  
Shallow Arches ◽  
Subsequent Deformation ◽  
Static Conditions

This paper discusses the response of long, shallow, elastic panels to uniform pressure loading. Under quasi-static conditions, the deformation of such panels is initially uniform along their length, and their response has the nonlinearity and instabilities characteristic of shallow arches. Shallower panels deform symmetrically about the midspan and exhibit a limit load instability. For less shallow panels, the response bifurcates into an unsymmetric mode before the limit load is achieved. A formulation and a solution procedure are developed and used to analyze the response of such panels beyond first instability. It is demonstrated in both cases that following the first instability the deformation ceases to be axially uniform and locqlizes to a region a few arch spans in length. A drop in pressure accompanies this localized collapse and causes unloading in the remainder of the panel. Subsequent deformation is confined to this region until membrane tension arrests the local collapse. Further deformation can occur at a constant pressure and takes the form of spreading of the collapsed region along the length of the panel. The lowest pressure at which this can take place (propagation pressure) can be significantly lower than the pressure associated with first instability.

Buckling Thresholds for Pre-Loaded Spherical Shells Subject to Localized Blasts

2019 ◽  
Vol 87 (3) ◽  
Author(s):  
Jan Sieber ◽  
John W. Hutchinson ◽  
J. Michael T. Thompson
Keyword(s):  
Impact Energy ◽  
Energy Barrier ◽  
Threshold Energy ◽  
External Pressure ◽  
Extensive Study ◽  
Energy Threshold ◽  
Integration Scheme ◽  
Spherical Shells ◽  
Buckling Mode ◽  
The Impact

Abstract This paper investigates the robustness against localized impacts of elastic spherical shells pre-loaded under uniform external pressure. We subjected a pre-loaded spherical shell that is clamped at its equator to axisymmetric blast-like impacts applied to its polar region. The resulting axisymmetric dynamic response is computed for increasing amplitudes of the blast. Both perfect shells and shells with axisymmetric geometric imperfections are analyzed. The impact energy threshold causing buckling is identified and compared with the energy barrier that exists between the buckled and unbuckled static equilibrium states of the energy landscape associated with the pre-loaded pressure. The extent to which the impact energy of the threshold blast exceeds the energy barrier depends on the details of its shape and width. Targeted blasts that approximately replicate the size and shape of the energy barrier buckling mode defined in the paper have an energy threshold that is only modestly larger than the energy barrier. An extensive study is carried out for more realistic Gaussian-shaped blasts revealing that the buckling threshold energy for these blasts is typically in the range of at least 10–40% above the energy barrier, depending on the pressure pre-load and the blast width. The energy discrepancy between the buckling threshold and energy barrier is due to elastic waves spreading outward from the impact and dissipation associated with the numerical integration scheme. Buckling is confined to the vicinity of the pole such that, if the shell is not shallow, the buckling thresholds are not strongly dependent on the location of the clamping boundary, as illustrated for a shell clamped halfway between the pole and the equator.

scholarly journals Buckling of spherical shells revisited

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160577 ◽  
Author(s):  
John W. Hutchinson
Keyword(s):  
Middle Surface ◽  
External Pressure ◽  
Maximum Pressure ◽  
Imperfection Sensitivity ◽  
Spherical Shells ◽  
Mode Localization ◽  
Axisymmetric State ◽  
New Findings ◽  
Post Buckling ◽  
The 1960S

A study is presented of the post-buckling behaviour and imperfection sensitivity of complete spherical shells subject to uniform external pressure. The study builds on and extends the major contribution to spherical shell buckling by Koiter in the 1960s. Numerical results are presented for the axisymmetric large deflection behaviour of perfect spheres followed by an extensive analysis of the role axisymmetric imperfections play in reducing the buckling pressure. Several types of middle surface imperfections are considered including dimple-shaped undulations and sinusoidal-shaped equatorial undulations. Buckling occurs either as the attainment of a maximum pressure in the axisymmetric state or as a non-axisymmetric bifurcation from the axisymmetric state. Several new findings emerge: the abrupt mode localization that occurs immediately after the onset of buckling, the existence of an apparent lower limit to the buckling pressure for realistically large imperfections, and comparable reductions of the buckling pressure for dimple and sinusoidal equatorial imperfections.

scholarly journals Nonlinear buckling behaviour of spherical shells: barriers and symmetry-breaking dimples

2017 ◽  
Vol 375 (2093) ◽  
pp. 20160154 ◽  
Author(s):  
John W. Hutchinson ◽  
J. Michael T. Thompson
Keyword(s):  
Symmetry Breaking ◽  
Large Range ◽  
External Pressure ◽  
Media Theory ◽  
Spherical Shells ◽  
Buckling Mode ◽  
Nonlinear Buckling ◽  
Accurate Evaluation ◽  
Post Buckling ◽  
Volume Controlled

The nonlinear axisymmetric post-buckling behaviour of perfect, thin, elastic spherical shells subject to external pressure and their asymmetric bifurcations are characterized, providing results for a structure/loading combination with an exceptionally nonlinear buckling response. Immediately after the onset of buckling, the buckling mode localizes into a dimple at the poles. The relations among the pressure, the dimple amplitude and the change in volume of the shell are determined over a large range of pole deflections. These results allow accurate evaluation of criteria such as the Maxwell condition for which the energies in the unbuckled and buckled states are the same and evaluation of the influences of pressure versus volume-controlled loadings. Non-axisymmetric bifurcation from the axisymmetric state, which occurs deep into the post-buckling regime in the form of multi-lobed dimples, is also established and discussed. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

On Establishing Buckling Knockdowns for Imperfection-Sensitive Shell Structures

2018 ◽  
Vol 85 (9) ◽  
Author(s):  
S. Gerasimidis ◽  
E. Virot ◽  
J. W. Hutchinson ◽  
S. M. Rubinstein
Keyword(s):  
Cylindrical Shells ◽  
Strong Dependence ◽  
Elastic Shell ◽  
External Pressure ◽  
Buckling Load ◽  
Spherical Shells ◽  
Nonlinear Buckling ◽  
Buckling Loads ◽  
Buckling Behavior ◽  
Global Buckling

This paper investigates issues that have arisen in recent efforts to revise long-standing knockdown factors for elastic shell buckling, which are widely regarded as being overly conservative for well-constructed shells. In particular, this paper focuses on cylindrical shells under axial compression with emphasis on the role of local geometric dimple imperfections and the use of lateral force probes as surrogate imperfections. Local and global buckling loads are identified and related for the two kinds of imperfections. Buckling loads are computed for four sets of relevant boundary conditions revealing a strong dependence of the global buckling load on overall end-rotation constraint when local buckling precedes global buckling. A reasonably complete picture emerges, which should be useful for informing decisions on establishing knockdown factors. Experiments are performed using a lateral probe to study the stability landscape for a cylindrical shell with overall end rotation constrained in the first set of tests and then unconstrained in the second set of tests. The nonlinear buckling behavior of spherical shells under external pressure is also examined for both types of imperfections. The buckling behavior of spherical shells is different in a number of important respects from that of the cylindrical shells, particularly regarding the interplay between local and global buckling and the post-buckling load-carrying capacity. These behavioral differences have bearing on efforts to revise buckling design rules. The present study raises questions about the perspicacity of using probe force imperfections as surrogates for geometric dimple imperfections.

A Comparison of Theoretical and Experimental Results From Spherical Shells With a Single Radially Attached Nozzle

1967 ◽  
Vol 89 (3) ◽  
pp. 333-338 ◽  
Author(s):  
F. J. Witt ◽  
R. C. Gwaltney ◽  
R. L. Maxwell ◽  
R. W. Holland
Keyword(s):  
Spherical Shell ◽  
Internal Pressure ◽  
Experimental Results ◽  
Strain Gages ◽  
Spherical Shells ◽  
Axial Thrust ◽  
Outside Diameter ◽  
Theoretical Results

A series of steel models having single nozzles radially and nonradially attached to a spherical shell is presently being examined by means of strain gages. Parameters being studied are nozzle dimensions, length of internal nozzle protrusions, and angles of attachment. The loads are internal pressure and axial thrust and moment loadings on the nozzle. This paper presents both experimental and theoretical results from six of the configurations having radially attached nozzles for which the sphere dimensions are equal and the outside diameter of the attached nozzle is constant. In some instances the nozzle protrudes through the vessel.

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