Plane Elasticity

Solving the full three-dimensional equations of elasticity can be incredibly complex for many practical applications. Fortunately, a large class of important engineering problems, such as those involving thin plates or very long uniform bodies, can be reasonably simplified to a two-dimensional analysis. This chapter is dedicated to the theory of plane elasticity, which provides the framework for these simplifications. We will explore the two primary idealizations used for this purpose: plane stress, which is suitable for thin bodies, and plane strain, which applies to long prismatic members. A significant portion of this chapter will be dedicated to a powerful mathematical tool known as the Airy stress function, which cleverly reduces the problem to solving a single governing equation. Finally, we will apply these concepts to solve several practical examples, demonstrating how this 2D framework is used to analyze real-world engineering structures.