Governing Differential Equation for the Deflection Curve of a Thin Bar — Lesson 1

This lesson covers the instability of rings, curved bars, and arches, which are fundamental components of versatile shell structures. The lesson delves into the derivation of the governing differential equation for the deflection curve of a thin bar with a circular center line. It also explores the deflection of a ring when subjected to two forces acting along the diameter. The lesson further discusses the concept of buckling under a uniformly distributed load and the derivation of the governing differential equation for the deflected shape of a curved bar. An illustrative example is provided to apply the derived equations to a practical scenario involving a ring subjected to radial load.

Video Highlights

00:43 - Derivation of the governing differential equation for the deflection curve of a thin bar with a circular center line
06:10 - Derivation of the change in curvature of the element
15:45 - Application of the derived equation to solve an example problem
26:36 - Explanation of the use of boundary conditions to determine the constants of the governing differential equation
31:03 - Use of Castigliano theorem to find the unknown moment M0

Key Takeaways

- The governing differential equation for the deflection curve of a thin bar with a circular center line can be derived using basic principles of physics.
- The deflection of a ring when subjected to two forces acting along the diameter can be calculated using the derived equation.
- The Castigliano theorem is a useful tool in finding the unknown moment in such structures.
- The lengthening and shortening of the diameter when the ring is subjected to point load can be calculated using the derived equations.