## 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  ## Linear Theory  ## 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  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.  ## Instability Patterns for various Buckling Load Factors.    ## Imperfections

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

• Geometry
• Support conditions
• Material ## 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 