# Second Order Buckling Analysis ## 07 Apr Second Order Buckling Analysis

For all linear analysis, the equilibrium of forces and moments is established on the undeformed structure, also called first order analysis. For a linear buckling analysis, this means that any deformation due to loads applied will not be considered, although in reality this deformation might significantly reduce the buckling load factor (BLF).

An example is a vertical beam with a central horizontal load. Although this load is resulting in a bending of the beam, the BLF as calculated by a linear buckling analysis is the same as for the beam without the horizontal load. However, in reality, the BLF of the bent beam would be lower than the BLF of the straight beam.

To get realistic results, the equilibrium of forces and moments has to be calculated on the deformed structure. This is also called large deformation analysis (https://fea-solutions.co.uk/large-deformation/), second order analysis or geometric non-linearity, which is one of the types of non-linear behaviour (https://fea-solutions.co.uk/non-linear-behaviour/).

In a second order buckling analysis, the loads are gradually applied over many small load steps. With each, the structure is deformed and the equilibrium of forces and moments is based on this newly deformed shape. Once the buckling load has been reached and the structure becomes unstable, the numerical solution will fail to converge and the analysis will stop. The effects that take place within the structure as it deforms are called second order effects. They are sometimes referred to as P-Δ and p-δ effects.

A second order buckling analysis is not a special analysis type, but just a stress analysis with large deformation. Hence, it results in the buckling load, the associated stresses and the deformation shape. No separate stress analysis is required. A second order buckling can also include other types of non-linear behaviour, like non-linear material (https://fea-solutions.co.uk/non-linear-material/) or non-linear contact (https://fea-solutions.co.uk/non-linear-contact/). However, with the added complexity associated with non-linear analyses, the model setup and computing power required is greater, which will lead to longer analysis run times.

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