Weighted Residual Methods in Structural Mechanics - I — Lesson 4

This lesson covers the fundamental concepts of weighted residual methods used in structural mechanics. It delves into the basis of the weighted residual method and different approximate methods used for solving governing differential equations. The lesson further explores methods like the Galerkin method, least square method, and collocation method. It also discusses how engineering analysis of problems like groundwater seepage, heat flow, and deformation of structural elements can be represented by differential equations. For instance, in complex structural mechanics problems, variational methods are used for deriving governing differential equations.

Video Highlights

00:42 - Discussion on the basis of the weighted residual method and different approximate methods used in structural mechanics
05:46 - Discussion on the concept of residue in the context of the weighted residual method
14:15 - Discussion on the concept of weight function in the weighted residual method
21:20 - Explanation of the Galerkin method where the weight function is the same as the trial function
29:03 - Explanation of the collocation method where the weight function is assumed such that the residue vanishes identically at n selected points

Key Takeaways

- The equilibrium approach, also known as the bifurcation approach, is used to estimate the critical load of structural elements.
- The energy approach is used when it's difficult to obtain the governing differential equation or find its exact solution.
- The imperfection approach helps clarify the discrepancy between theory and experimental results, especially in shell buckling.
- The total potential energy of a system is found in the bent configuration and is expressed as the strain energy stored due to bending minus the external work done.
- The energy approach also provides insights into the characteristics of equilibrium through a plot between total potential energy and displacement parameter.