Question

The closed thin-walled rectangular channel shown in figure (i) is opened by introducing a sharp cut at the center of the bottom edge, as shown in figure (ii). Which one of the following statements is correct?

• A. Centroids of (i) and (ii) coincide while shear centers do not
• B. Shear centers of (i) and (ii) coincide while centroids do not
• C. Both centroids and shear centers of (i) and (ii) coincide
• D. Neither centroids nor shear centers of (i) and (ii) coincide

Hint:

When analyzing thin-walled structures with cuts or holes, remember that while centroids are unaffected by cuts in symmetric sections, shear centers often shift because of changes in the shear stress distribution.

Answer 1

The Correct Option is A

In this problem, we are concerned with the centroid and shear center of a thin-walled rectangular channel, both of which are fundamental concepts in the mechanics of materials. Let's break down the problem and the impact of introducing the sharp cut at the center of the bottom edge. 1. Centroid: The centroid of a thin-walled structure is determined solely by the geometry of the boundary and is the point about which the shape is symmetrically distributed. For both the original (i) and the modified (ii) configurations, the overall shape and area distribution along the vertical axis remain symmetric, even though a sharp cut is introduced. Therefore, the centroid of the open channel (ii) remains coincident with the centroid of the closed channel (i), as no material has been removed in a way that alters the overall geometric distribution of the section. Hence, the centroid of (i) and (ii) coincide. 2. Shear Center: The shear center is the point at which an applied shear force does not cause any twisting of the structure. In the closed thin-walled channel (i), the shear center coincides with the centroid due to the symmetry and continuity of the section. However, once a sharp cut is introduced at the center of the bottom edge, the symmetry of the section is broken. The flow of shear stress is altered, and the shear center of the open section (ii) no longer coincides with the centroid. This is because the cut disrupts the uniform shear flow, causing the shear center to shift outside the section. Therefore, the centroids of (i) and (ii) coincide, but their shear centers do not.