3.2. Circular cross-section

In the derivation for rectangular beams, we had assumed that τ is independent of z coordinate. For a circular beam however, this assumption cannot be used. Consider the beam shown in Figure 10 and analyze one of its cross-sections.

When dealing with circular cross-sections, the assumption that shear stress (τ) does not vary with the z-coordinate (the direction perpendicular to the plane of the cross-section) is not valid. Unlike rectangular beams where this assumption simplifies calculations, in circular beams, shear stress is influenced by variations through the thickness of the beam. Therefore, the behavior of shear stress must be analyzed differently.

Think of a pizza. If you slice through it, the cheese and toppings align differently towards the edges compared to the center. This reflects how the shear stress distribution changes when analyzing circular beams. Just as the pizza's topping distribution isn’t the same from the center to the edge, the shear stresses in a circular beam vary depending on the location within the cross-section.

If τ is constant along lines parallel to the z-axis, the shear stress will be as shown in Figure 11. Basically, it is non-zero even at the ends.

In our analysis, if we assume that shear stress remains constant when moving along the z-axis, we can observe from Figure 11 that shear stress does not become zero at the ends of the circular cross-section. This contradicts the assumption because at the lateral surface (the outer edge of the circular cross-section), the shear stress must actually be zero. Therefore, a constant assumption leads to incorrect conclusions about the distribution of shear stress.

Imagine holding a tight rubber band around a round balloon. If you pull the rubber band evenly at the top and bottom, it doesn't just slip off the sides; instead, it applies pressure all around the surface. Translating this to shear stress, if you assume that it stays constant even at the surface of a circular beam, you're misinterpreting how stresses distribute themselves under load.

Let us work with a cylindrical coordinate system and assume a radial distributed load is acting (e.g., pressure load) which could also be zero. In that case, τ must be zero on the lateral surface. So, τ must also be zero along the periphery of the cross section.

When a radial load is applied to a circular beam, this pressure affects the shear stress distribution. Specifically, if there is a radial load exerted, the shear stress acting on the lateral surface of the beam must be zero. If it were not, this would imply that there is a radial component of shear stress at the edge of the cross-section, which is physically impossible in a circular cross-section given that the load is being applied radially. Thus, the understanding of shear stress behavior is critical in designing these structures.

Consider filling a round balloon with air. The pressure you exert from the inside pushes out uniformly, but the balloon material doesn’t stretch inward at the surfaces where it’s elastic. Similarly, in a circular beam, the internal stresses must balance the external loads without creating internal shear at the outer edge.

However, if we look at Figure 11, the assumption of τ being independent of the z coordinate leads to a non-zero radial component which is a contradiction. Thus, we can conclude that for circular cross-sections, considering τ independent of z is not a good assumption. Still, this assumption is often used since it gives an approximate distribution of shear stress.

The conclusion drawn is that assuming shear stress does not vary along the z-coordinate in circular cross-sections can lead to contradictions. Although this assumption might simplify the calculations for some applications, it does not accurately reflect the real behavior of shear distributions. Thus, while it can be useful for preliminary designs or rough estimates, more rigorous models are needed for precise calculations.

Think of how a simplified map shows geographic features without detail — it provides a rough idea but misses nuances essential for navigation. Similarly, using the assumption that shear stress is independent of z can guide initial designs but may fail under precise engineering requirements where real-life conditions matter.

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