1.1. Introduction

In case of pure bending, we had the same moment acting on every cross-section. For this reason, pure bending is also called uniform bending. We will now move on to non-uniform bending of a beam where bending moment is not uniform along the length of the beam.

This chunk introduces the concept of non-uniform bending as a progression from pure bending. In pure bending, the bending moment is consistent across the entire beam, resulting in a uniform distribution of stress and curvature. Non-uniform bending, on the other hand, occurs when different parts of the beam experience different moments, leading to varying curvature. This concept is essential to grasp because it sets the context for understanding how different loads affect the beam's structure.

Imagine a long, flexible ruler. If you bend it evenly along its length, it will bow uniformly (pure bending). However, if you press down harder in the middle while holding the ends, the middle will flex more, resulting in a non-uniform curve. This scenario is akin to non-uniform bending in beams.

In the previous lecture, we derived the following relation for bending moment M and curvature κ: M = EIκ. If bending moment M is not constant along the length, bending curvature will also not be constant.

The equation M = EIκ relates the bending moment (M), the modulus of elasticity and moment of inertia (EI), and the curvature (κ) of the beam. In cases of non-uniform bending, as the bending moment changes, so too does the curvature. This shows that the way a beam bends is intrinsically linked to how forces are applied along its length.

Consider a basketball hoop. If someone pushes down harder on one side of the hoop, that side will flex more than the other. Thus, parts of the hoop will experience different curvatures, much like how a beam behaves under uneven loading.

We now consider another case of loading where we can also have distributed load b(x) acting on the beam. The load b(x) is assumed to act in +y direction.

A distributed load means that the force is spread out over a length of the beam rather than applied at a single point. This kind of loading can complicate the bending behavior of the beam because different sections experience variable forces. For engineering applications, understanding how to calculate the impact of these distributed loads is crucial for ensuring the structural integrity of beams in buildings and bridges.

Think of a flat shelf loaded with books. If you place all the books in the middle, the shelf will bend down more in that area compared to the ends where there are no books. This variation in loading across the length of the shelf is similar to a distributed load on a beam.

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