## 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

## 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
- Load
- Support conditions
- Material

## Non – Linear Buckling Analysis.

- More accurate approach for finding the structures 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