For the hollow elliptical section in Example 9.5, explicitly show that the given stress function (9.6.1) will

Question:

For the hollow elliptical section in Example 9.5, explicitly show that the given stress function (9.6.1) will yield the constant value specified by (9.6.2) on the inner boundary. Next using the methods from Example 9.1, show that the general torque relation (9.3.31) will also produce the result (9.6.3). Finally verify the maximum shear stress relation (9.6.4).

Data from example 9.5

Consider the torsion of a bar with a hollow elliptical section as shown in Fig. 9.15. The inner boundary is

x (ka) + y (kb) = 1 y Xx

and this form satisfies the governing equation, boundary conditions, and the multiply connected condition

Equation 9.6.1

e = ab' at b

Equation 9.6.2

; = a?b? al + bl 52 (4-1)

Equation 9.3.31

T = 2 | | pdxdy+20A JR

Equation 9.6.3

T= b a+b (ka)3(kb)  (ka)2 + (kb)2 _a? + bab (1 - kt)  2

Equation 9.6.4

Tmax 2T 1 abi - k4

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