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**Hollow elliptical section**

Here we study the torsion of a bar with a hollow elliptical section as shown in figure 9.23 where the inner boundary is a scaled ellipse similar to that of the outer boundary. Since, the inner ellipse is a scaled outer ellipse, the Prandtl stress function obtained for solid ellipse cross section in section 9.3.1 is a constant at the inner surface. Hence, the boundary condition (9.118) holds and therefore this stress function

is a solution to this boundary value problem of twisting of a hollow elliptical bar. Now, the constant C_{1}, the value of Ï† at the inner boundary is obtained as,

Using (9.127) and (9.128) in (9.126), twisting moment is obtained as

Since, the stress function is same as in the case of a solid elliptical section, the stress distribution is also identical as before. Therefore, the maximum shear stress occurs at x = 0, y = Â±b and is given by,

Before concluding this section we observe that this solution scheme can be applied to other cross sections whose inner boundary coincides with a contour line of the stress function in the corresponding solid section problem.

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