Theory of Structures II (ENCE 252): Theorem of Displacements

Understanding the Theorem of Displacements

This chapter delves into the Theorem of Displacements, a critical area in structural analysis that focuses on calculating the deflections and rotations of structural members under load. While the previous chapter established how to classify structures, this chapter provides powerful energy-based methods to quantify their behavior.

The cornerstone of this topic is Castigliano’s Theorem, which relates the partial derivative of a structure’s total strain energy to its displacement. This elegant principle allows engineers to determine the deflection at any point in a structure by applying a virtual force. For example, Castigliano’s first theorem states that the displacement $\delta_i$ at a point $i$ is the partial derivative of the total strain energy $U$ with respect to the force $P_i$ applied at that point:

$\delta_i = \frac{\partial U}{\partial P_i}$

Mastering these concepts is essential for analyzing statically indeterminate structures, where equilibrium equations alone are insufficient. The principles learned here form the basis for more advanced topics and are fundamental to ensuring that structures are not only strong but also sufficiently stiff to meet serviceability requirements.