This lesson covers the analysis of columns with unspecified boundary conditions. It delves into the critical load of columns with different boundary conditions using the 4th order differential equation, which is the strong form of the governing differential equation. The lesson further explores the application of this methodology to any column, regardless of the boundary conditions. It also discusses the concept of elastically supported ends, which are common in most structural configurations. The lesson concludes with the derivation of the critical load for a column with elastically supported ends. For instance, it explains how a cantilever column supported at the end by a spring that is elastically restrained can be solved to find the critical load and mode shape.

- The 4th order differential equation can be used to analyze columns with any type of boundary conditions.
- Columns in most structural configurations are supported by other structural elements providing elastic type of restraint at the ends.
- The critical load of a column with elastically supported ends can be derived using the 4th order differential equation.
- The solution to the problem involves finding the critical load (P critical) and the mode shape.
- The best way to solve the final equation is by graphical method or trial and modification.

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