20.1.1. Flexure

Fig.20.1 shows portion of an originally straight beam which has been bent to the radius $\rho$ by end couples $M$, thus the segment is subjected to pure bending.

Pure bending occurs when a straight beam is subjected to forces (couples) at its ends, causing it to bend into an arc shape. In this situation, the internal forces distribute uniformly across the cross-section, leading to predictable stress and strain patterns.

Think of a bent paper clip. When you apply a force at both ends, it bends smoothly. Similarly, a beam subjected to bending forces behaves in a controlled manner, allowing engineers to predict how it will respond to loads.

It is assumed that plane cross-sections normal to the length of the unbent beam remain plane after the beam is bent.

This assumption implies that as the beam bends, the shapes of the cross-sections do not distort; they merely move and rotate. This is crucial in structural analysis because it simplifies the calculations related to stress and strain in the beam.

Imagine a stack of pancakes. When one is slightly tilted, the others remain flat and level. Similarly, in a bent beam, each cross-section is still flat like a pancake, even though the beam itself has changed shape.

Therefore, considering the cross-sections AB and CD a unit distance apart, the similar sectors Oab and bcd give $y = \frac{\rho (b-c)}{d}$. Where $y$ is measured from the axis of rotation (neutral axis). Thus strains are proportional to the distance from the neutral axis.

In a beam under bending, the strain experienced at any point varies depending on its distance from the neutral axis, which is the line along the beam where no elongation occurs during bending. The further away from this axis, the greater the strain (and consequently the stress). This relationship is crucial for determining how much load a beam can safely handle.

Think of a rubber band stretched from both ends. The part of the band in the middle (neutral axis) stays at its original length, while the sections on either side get stretched more. Just like the strain in the rubber band increases the further you move from the center.

The corresponding variation in stress over the cross-section is given by the stress-strain diagram of the material rotated 90° from the conventional orientation, provided the strain axis $\epsilon$ is scaled with the distance $y$.

As strain varies, so does stress. By referencing the stress-strain relationship, engineers can determine how much stress each section of the beam can handle. When the beam bends, the stress distribution across the cross-section becomes non-uniform, with the highest tensile stress on one side and the highest compressive stress on the other.

Consider a tightly held rope. While it’s pulled from both ends, the middle remains unchanged, but the ends experience different levels of stress based on how much they are being jerked or twisted. Similarly, a bent beam exhibits varying stress levels across its section.

The bending moment M is given by $M = \int y \sigma dA$, where $dA$ is an differential area a distance $y$ from the neutral axis. Thus the moment M can be determined if the stress-strain relation is known.

To compute the bending moment, engineers integrate the product of the distance from the neutral axis $y$ and the stress at that distance $\sigma$ over the entire cross-sectional area. This equation is fundamental in understanding how external loads applied to the beam translate into internal moments that cause bending.

Think of balancing a seesaw with a child on one end. The moment created is a product of how far the child is from the pivot point (y) and their weight (stress). The farther they are, the greater the moment, just like how a bending load creates moments in the beam.

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