3. Non-uniform bending of unsymmetrical cross-sections

In non-uniform bending, a shear force is also present in the cross section due to which the bending moment varies along the length of the beam. The direction of neutral axis will again be governed by equation(11).

In non-uniform bending of beams, both shear forces and bending moments are considered. This means that instead of a constant turning effect, the bending moment varies along the length of the beam. This variation requires a reevaluation of the neutral axis, previously defined in simpler scenarios where only uniform loading was considered. Here, the neutral axis's direction is still determined by a certain equation from previous sections, ensuring that we account for this complexity.

Think of a long, flexible rubber band. If you bend it in one place (apply a moment), the curve might not be the same along its entire length due to how you hold and stretch it—this is akin to non-uniform bending where forces act differently along the beam's length.

For symmetrical beams, for simplicity, we had assumed uniform shear stress distribution on lines parallel to neutral axis. Will such an assumption be valid for unsymmetrical beams also? It turns out that we cannot make such an assumption.

In symmetrical beams, we often assume shear stress is evenly distributed along lines parallel to the neutral axis; however, for unsymmetrical beams, this is not valid. This is due to how shear forces react across non-uniform cross-sections. When forces are applied, the shear stress does not follow a simple pattern, particularly near the edges of the beam where side effects come into play.

Imagine buttering a slice of bread—when you apply pressure near the edges of the slice, the butter might smear unevenly compared to when you press in the center. Similarly, shear stress in unsymmetrical beams does not distribute evenly across the entire section.

However, we do obtain analytical results for special types of cross-sections - they have to be thin and open which we discuss now.

Despite the complexities, specific types of cross-sections—particularly thin and open ones—permit analytical solutions. This means that while general shapes might not yield easy calculations for stress and bending, these specific shapes allow us to derive formulas that can predict performance reliably. The reason behind this lies in their geometry and how stress distributes across them.

Consider a piece of paper made into a thin tube. It can bend and flex easily compared to a thick block of wood. The tube's thin and open nature allows for predictable bending behaviors in response to stress, just like our analytical solutions.

For thin cross-sections, however, the thickness is so small that all points can be safely assumed to lie near the periphery.

In thin cross-sections, we assume that every point lies near the outer edge of the beam. This simplifies our analysis as the shear stress distribution can be approximated to be consistent across that small thickness. This homogeneity in distribution eases the complexity typically faced in non-uniform cross-sections.

Think of a thin sheet of plastic—when you apply pressure, the stress is felt uniformly along the surface because the thickness is negligible. Thus, we can easily analyze how it will react compared to something much thicker.

As the shear flows from one end to the other and that it is uniform through the thickness, one can parameterize its distribution using arc-length coordinates along the cross-section’s periphery.

When analyzing how shear flows through thin cross-sections, we can set a coordinate system along the edge (or periphery) of the beam. This method allows us to track how shear stress varies as we move around the cross-section. The assumption of uniform shear across the tiny thickness simplifies calculations and helps create formulas that can predict the shear stress accurately.

Imagine drawing a line around a donut and analyzing how icing flows around it as you rotate the donut. By just observing the outline, you can predict how the icing thickness changes, similarly, we analyze shear flow along the cross-section’s boundary.

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