 By now you are very familiar with the linear shear stress distribution in a solid circular shaft subjected to torsion as pictured here. But I have a problem with this shaft. Part of it is not pulling its own weight. The magnitudes of the shear stresses and moment arms in the center of the shaft are small compared to that of the outer portions of the shaft, resulting in the majority of the torque being carried by the outer regions. So for weight efficiency it is better to eliminate this lazy portion of the shaft and use a hollow cross-section. We already know that for such a hollow cross-section the same linear stress distribution of a solid shaft also applies when the shaft is circular. But what if the wall thickness of the shaft is very small compared to its radius? As it turns out there are some simplifications that can be made which will make our lives a lot easier. For a shaft with a wall thickness much smaller compared to its radius the variation in shear stress and moment arm become very small through the wall thickness. It is thus convenient to approximate this stress state as constant through the wall thickness. To help remember that we are making this simplification it is convenient to multiply the stress by the wall thickness resulting in a shear force per unit length around the perimeter of the shaft. This quantity represents the flow of the shear force around the cross-section and is thus referred to as the shear flow. For the thin walled circular shaft shown here it is clear due to the constant radius and shaft wall thickness that the shear flow will remain constant. But what if this cross-section is not circular and what if the thickness around the perimeter is variable? To generalize our analysis we will consider an arbitrary thin walled shaft with a closed cross-section subjected to torsion. We will allow the thickness to vary around the cross-section however the thickness at every location must be much smaller than the distance of the cross-section perimeter to the central axis of the shaft. To illustrate this constraints consider the following three cross- sections. The first section clearly meets all of these constraints however in the rectangular section the wall thickness is not sufficiently small compared to the overall width of the cross-section and thus would not be suitable for this analysis. Finally the last cross-section is also not suitable as it is an open section rather than a closed section. Let's use the first section to illustrate our generalized shaft. If we take a closer look at our generalized shaft we can analyze a rectangular element passing through the entire thickness of the shaft wall. In this shaft the thickness can vary along the perimeter but it is constant along the length of the shaft due to the constant cross-section. If we zoom in on this element and examine the shear flow distribution we can see that along BC we will allow the shear flow to vary from QB at point B to QC at point C. As the cross-section of the shaft is constant the thickness along length AB is constant and equal to the thickness at B. Combined with the principle of complementary shear stress this means that the shear flow along length AB and the shear flow at A along AD are both equal to QB. Repeating the same for the bottom half of the element we obtain that the shear flow along length DC is equal to QC. With the shear flows defined we can now enforce equilibrium along the length of the shaft. Looking at the forces acting in the direction of the axis of the shaft we find that the resultant shear force QB times DX along length AB is equal to the resultant shear force QC times DX along length DC. Simplifying we find that QB must equal QC and thus the shear flow is constant around the perimeter of the section. So shear flow is constant around the perimeter of a closed thin walled shaft subjected to torsion. This is quite handy and will result in a simplified torsion formula for thin walled shafts but don't confuse a constant shear flow with a constant shear stress. Shear flow is the product of the average shear stress through the thickness with the wall thickness itself. For a constant shear flow if the wall thickness increases the shear stress has to decrease in order to maintain the same flow. This is analogous to a constant flow rate through a variable section pipe. If the pipe cross section increases the velocity of the fluid decreases to maintain the same flow rate.