315_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

315_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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Unformatted text preview: Sec_3.8.qxd 9/27/08 1:13 PM Page 309 SECTION 3.8 Statically Indeterminate Torsional Members 309 Solution 3.8-9 Bar with a hole IPA Polar moment of inertia at left-hand end IPB Polar moment of inertia at right-hand end fB TB(L/2) TB(L/2) + GIPB GIPA T0(L/2) GIPA L 50 in. L/2 25 in. d2 outer diameter L d 2IPA x L + IPB 2IPB diameter of hole COMPATIBILITY x Torque applied at distance x Find x so that TA ‹ IPB TB TB T0 ‹ TA 0 B 3L 4IPB L 4IPA soLVE FOR x: EQUILIBRIUM TA T0 L L c + G 4IPB 4IPA fB 2.4 in. T0 (2) Substitute Eq. (1) into Eq. (2) and simplify: 3.0 in. d1 L/2) T0(x GIPB TB T0 2 (1) REMOVE THE SUPPORT AT END B x IPB IPA x L a3 4 IPB b IPA 4 d2 4 d1 4 d2 1 d1 4 L c2 + a b d 4 d2 a d1 4 b d2 ; SUBSTITUTE NUMERICAL VALUES: x B 2.4 in. 4 50 in. c2 + a bd 4 3.0 in. 30.12 in. ; Angle of twist at B Problem 3.8-10 A solid steel bar of diameter d1 25.0 mm is enclosed by a steel tube of outer diameter d3 37.5 mm and inner diameter d2 30.0 mm (see figure). Both bar and tube are held rigidly by a support at end A and joined securely to a rigid plate at end B. The composite bar, which has a length L 550 mm, is twisted by a torque T 400 N m acting on the end plate. (a) Determine the maximum shear stresses 1 and 2 in the bar and tube, respectively. (b) Determine the angle of rotation (in degrees) of the end plate, assuming that the shear modulus of the steel is G 80 GPa. (c) Determine the torsional stiffness kT of the composite bar. (Hint: Use Eqs. 3-44a and b to find the torques in the bar and tube.) Tube A B T Bar End plate L d1 d2 d3 ...
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