Stability Analysis of Imperfect shells

General

  • Circular cylindrical shells are very stiff and light structural elements
  • If a shell is loaded in such a way that the membrane energy can be converted in bending energy, it may fail dramatically.
  • Very large deflections are required to transform the membrane energy to bending energy

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Linear Theory

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linear2

Linear Buckling Analysis

  • Predicts the buckling strength of an ideal elastic structure
  • The results are the buckling load factors that scale the load applied in static analysis
  • A structure can have many buckling load factors, each of them corresponds to a deformation pattern (buckling mode shape)
  • The lowest load factor is of interest
  • A structure becomes unstable when a load reaches its buckling value

 

 

linearBuckling

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Critical load factor for the perfect system

  • Pcrit = 1890.4  N/mm

Max Displacement

  • u = 1.011 mm

However: The critical buckling load of an imperfect shell is dramatically reduced compared to the ideal structure.

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Instability Patterns for various Buckling Load Factors.

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Pcrit4

Pcrit2

Pcrit3

Imperfections

Linear buckling predicts the theoretical buckling strength of a perfect  structure. However, non linearities reduce dramatically the buckling load.

  • Geometry
  • Load
  • Support conditions
  • Material

imperfections

Non – Linear Buckling Analysis.

  • More accurate approach for finding the structure’s limit load
  • The load is increasing gradually up to the point at which the structure becomes unstable.
  • Imperfections are added by updating the geometry from the linear buckling analysis
  • Hence, we add the displacements of the mode shapes reduced by a scaling factor

Steps for Nonlinear Buckling Analysis

  • In the first run, a convergence error occurs because the structure becomes unstable
  • Inspection of the Ansys monitor file. Buckling may have started to occur if:
    • The program bisects the load step increment and attempts a new solution at smaller load
    • The maximum displacement has an instantaneous change in value
    • The maximum displacement has an instantaneous change in sign
  • We locate the corresponding time
  • To be sure that we have reached the critical point, we start a postbuckling analysis
  • Because postbuckling analysis is unstable we apply stabilization using energy method
  • We control the stabilization using energy ratios
  • Higher energy ratios result in convergence but the system becomes stiffer.
  • Smaller energy ratios result in no convergence.

Linear Buckling Analysis -– Nonlinear Buckling Analysis

  • The critical load factor from the nonlinear buckling analysis is much lower than that of the linear buckling analysis
  • The cylinder buckles faster because of the geometric imperfections that were induced in the model

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