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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?

(i) a closed thin-walled rectangle 2a wide and b high; (ii) the same rectangle with a cut at the middle of the bottom edge, leaving two lengths a

  • 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

Answer: A

The centroids coincide, while the shear centres do not.

A sharp cut removes a negligible amount of wall, so the material of the section, and hence its centroid, is unchanged. The closed rectangular tube is doubly symmetric, so its shear centre lies at the centroid. Cutting the bottom wall at its centre leaves an open section that is still symmetric about the vertical axis, so its shear centre lies on that axis, but the shear flow now has to start from zero at the cut. The resulting shear flow distribution is different, and the shear centre of the open section moves along the axis of symmetry, away from the centroid.

So the options saying the shear centres coincide (with or without the centroids) are wrong, and so is the option saying the centroids move.

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