Shear stress in circular shafts changes based on how forces are applied to them. The main forces that affect this shear stress are twisting (torsion), pushing or pulling along the length (axial loads), and forces from the side (lateral loads).

1. Torsion: When a shaft is twisted, the shear stress is evenly spread out across its cross-section.

   The formula to find the shear stress ($\tau$) at a distance $r$ from the center is:

   $\tau = \frac{T r}{J}$

   Here, $T$ is the twisting force (torque) and $J$ is a measure of how much the shaft resists twisting.

   For solid circular shafts, we calculate $J$ using:

   $J = \frac{\pi d^4}{32}$

   This shows that if the diameter (d) is larger, $J$ also gets bigger. This means that the shear stress will be lower for the same twisting force.

2. Axial Load: If we push or pull on the shaft with an axial load ($P$), it causes a type of stress called normal stress ($\sigma$) that spreads evenly over the area ($A$):

   $\sigma = \frac{P}{A}$

   To find the area ($A$), we use:

   $A = \frac{\pi d^2}{4}$

   Even though axial loads mainly create normal stress, they can also lead to some extra shear stresses because of changes in shape or bending, especially in thin shafts.

3. Lateral Load: Lateral loads cause the shaft to bend, which changes how shear stress is spread.

   The shear stress ($\tau$) at a distance $y$ from the center when the shaft is bending is calculated using:

   $\tau = \frac{VQ}{Ib}$

   In this formula, $V$ is the force inside the shaft, $Q$ is a measure of the area related to the center, $I$ is another measure of how the shaft bends, and $b$ is the width of the shaft.

   The highest shear stress is at the center and decreases as you move outwards.

In summary, different types of loads affect how shear stress is spread in a shaft. Engineers need to understand these effects when designing safe and effective structures.