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? ? ? ? ?  ? ? Problem Solution – Homework #3 3.1 Solution : 1. Saint-Venant: as the shaft twists the plane, cross-sections are warped but the projections on the x-y plane rotate as a rigid body, then, (3.1.1) : warping function θ: angle of twist per unit length of the shaft, very small. 2. From the displacement field above, it is easy to obtain that Stress-strain relationship: Equilibrium equations: This equation is identically satisfied if the stresses are derived from a stress function so that (3. 1. 2) 3. From the displacement field and stress-strain relationship, we can obtain (3. 1. 3) (3. 1. 4) Compatibility equation: Prandtl stress function (3. 1. 5) 4. Boundary conditions,

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But for solid sections with a single contour boundary, this constant can be chosen to be zero. Then we have the boundary condition on the lateral surface of the bar. 5. For a bar with circular cross-section, assume the Prandtl stress function as Substitute into (3.1.5), we obtain Using (3.1.2), we have Comparing with (3.1.3) and (3.1.4), we have Hence, we conclude . This means that the cross-section remains plane after torsion. In other words, there is no warping. Therefore can be verified, and it successfully expresses the statement. 3.2 Solution : 1. To show that the Prandtl stress function for bars of circular solid sections is also valid for bars of hollow circular sections, we have to show that the Prandtl stress function for hollow circular sections satifies equilibrium equations, compatibility equations as well as traction boundary conditions. a. Equilibrium equations Prandtl stress functions by their definition must satify equilibrium equations. b. Compatibility equations Use the Prandtl stress function as it stated for bars of circular solid sections Here we use . Assuming
would be fine too. Then substitute into (3.2.5), we have Therefore we have a stress function for bars of hollow circular sections satisfying the compatibility equation c. Traction boundary conditions To satisfy the traction boundary conditions we must show on the traction free surfaces, It shows that the B.C.’s have been satisfied. d. Since equilibrium equations, compatibility equations and traction boundary conditions are all satisfied, the Prandtl stress function for bars of circular solid sections is also valid for bars of hollow circular sections. 2. Compare torsion constant a. The torque produced by the stresses is (3. 2. 7) Substituting (3.2.6) into (3.2.7) and use polar coordinates to perform integration, we have, Comparing with , we have the torsion constant b. Using (3.59) in the textbook for thin-walled sections, we have the approximate torsion constant

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where is the area enclosed by the centerline of the wall section. 3.
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êµ¬ì¡°ì—­í•™ë°_ì‹¤í

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