This lesson covers the derivation of the governing differential equation for columns with unspecified boundary conditions. It explains how to find the critical load of a column with hinged-hinged boundary conditions using the fourth-order differential equation. The lesson then moves on to discuss other sets of boundary conditions, such as fixed-fixed and fixed-free columns, to determine the Euler buckling load. The methodology for deriving the strong form of the governing differential equation and the solution methodology for finding the critical load is also discussed. For instance, the lesson explains how to use boundary conditions to derive equations and solve them to find constants in the general solution. The lesson concludes with the calculation of the critical load and mode shape for different boundary conditions.

- The governing differential equation for columns with unspecified boundary conditions can be derived.
- The critical load of a column with hinged-hinged boundary conditions can be found using the fourth-order differential equation.
- Other sets of boundary conditions, such as fixed-fixed and fixed-free columns, can be considered to determine the Euler buckling load.
- The strong form of the governing differential equation can be derived using the same methodology.
- The critical load and mode shape for different boundary conditions can be calculated using boundary conditions and the general solution.

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