Explain the pure torsion of thin walled bars of open cross section.
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Answer:
Chapter 9
End Torsion of Prismatic Bars
9.1 Overview
In the last chapter, we studied the response of straight prismatic beams subjected to bending moments. In this chapter, we study the stresses that develop and the displacement that these members undergo when subjected to twisting moment. While in the analysis of beams subjected to bending moments we maintained some generality with the loading, the study of torsion is focused on a single loading case - one end fixed and the other end free to twist, as shown in figure 9.1. Here the double arrow indicates the twisting moment. However, we study this boundary value problem for a variety of cross sections. This analysis for torsion depends on whether the section is thick walled or thin walled as in the case of bending moments. It also depends on whether the section is open or closed. We shall discuss in detail as to when a section is closed subsequently.
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Figure 9.1: Schematic of a straight prismatic member subjected to end torsion
Before proceeding further, we would like to point out the change in the orientation of the coordinate system, from that used in the study of the beams. This is necessitated because for some problems we would be using cylindrical polar coordinates, and want the cross section to be in the xy plane. We use cylindrical polar coordinates with this orientation so that the boundary of the cross section can be defined easily. Since, the value of displacement or stress in a material particle is independent of the coordinate system used, it is expected that the analyst pick coordinate systems that is convenient for a problem and not that which he is comfortable with. Hence, the change.
For this orientation of the coordinate system, following the same steps as discussed in detail in chapter 8, it can be shown that the torsional moment, Mz is given by,
∫
Mz = (σyzx - σxzy)dxdy,
a
(9.1)
and for completeness, the other two components of the moment, which are bending moments are,
∫
My = - σzzxdxdy, (9.2)
∫ a
M = σ ydxdy. (9.3)
x a zz
Comparing equations (9.1) with (9.2) and (9.3) it can be seen that while the bending moments are due to normal stresses, torsional moment is due to shear stresses. This is the major difference between the bending and twisting moment. While the bending moment gives raise to normal stresses predominantly, twisting moment gives raise to shear stresses predominantly.
Now let us see how to classify the sections. As in case of bending, a cross section would be classified as thin walled if the thickness of the cross section is such that it is much less than the characteristic dimensions of the cross section. Typically, if the ratio of the thickness of the cross section to the length of the member is less than 0.1, the section is classified as thin. If the section is not thin, it is considered to be thick.
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Figure 9.2: Section classified as closed sections when subjected to torsion
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Figure 9.3: Section classified as open sections when subjected to torsion
If a section when twisted can deform such that plane sections before deformation remain plane after twisting is called closed section. Sections which do not deform in the above manner are called open sections. Solid Circular cross section is an example of closed section and solid rectangular cross section is an example of open sections. Similarly, a thin walled annular cylinder (see figure 9.2) is an example of closed section and if the same annular cylinder has a longitudinal slit (see figure 9.3), it is an example of open section. All closed section allows for continuous variation of shear stresses such that it is no where zero except at the centroid of the cross section. A section which requires the shear stresses to be zero at locations apart from the centroid of the cross section, like at the corners of a rectangular cross section, is an open section. It should be pointed out that in elliptical shaped cross section also the shear stress is zero only at the centroid of the cross section but is an open section as plane sections do not remain plane after twisting. Thus, the requirement on the shear stress is just necessary but not sufficient to classify a given section as closed.
In the following section, we find the displacement field and stress field for twisting of a thick walled sections and then focus on thin walled sections.