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1 WORCESTER POLYTECHNIC INSTITUTE MECHANICAL ENGINEERING DEPARTMENT STRESS ANALYSIS ES-2502, C 2012 Lecture 17: 10 February 2012
2 General information Instructor: Cosme Furlong HL-151 (508) Teaching Assistants: Morteza Khaleghi HL-150 (508) Tatiana Popova
3 Torsion: shear strains Cross section max c Shear strains vary linearly within a section: ( ) c max
4 Torsion formula According to Hook s law (linear elasticity): ( G ) Shear stresses also vary linearly within a section: ( ) c max Differential Force: dt df da Differential Torque: ( da)
5 Torsion formula Integrating torque: T A ( da) A c max da max 2 c A da Define: A 2 da Polar area moment of inertia max Torsion formula for stresses: (linear elastic) T c and ( ) T
6 Torsion formula: solid circular bar Linear variation of shear stress
7 Torsion formula: polar area moment of inertia Solid bar A 2 0 c 2 da ( 2 d) Solid, circular, section: c d c c 0
8 Torsion formula: polar area moment of inertia Tubular bar c i c o A 2 da c c i o 2 (2 d ) c o 2 d 2 ci c c o i Tubular section: 4 ( c o c 2 i 4 )
9 Torsion formula: tubular bar Linear variation of shear stress
10 Torsion: example A A shaft is made of a steel alloy having an allowable shear stress of allow = 12 ksi. If the diameter of the shaft is 1.5 in., determine the maximum torque T that can be transmitted. What would be the maximum torque T if a 1-in. diameter hole is bored through the shaft? Sketch the shear-stress distribution along a radial line in each case.
11 Torsion: example B The solid shaft is fixed to the support at C and subjected to the torsional loadings shown. Determine the shear stress at points A and B and sketch the shear stress on volume (stress) elements located at these points.
12 Torsion: example C The solid 30 mm diameter shaft is used to transmit the torques applied to the gears. Determine the absolute maximum shear stress on the shaft.
13 Torsion: example D The coupling is used to connect the two shafts together. Assuming that the shear stress in the bolts is uniform, determine the number of bolts necessary to make the maximum shear stress in the shaft equal to the shear stress in the bolts. Each bolt has a diameter d.
14 Power transmission P T with: rad sec 2 f f Hz T, SI: FPS: 1W 1hp 1N 550 m s ft lb s
15 Torsion: example E The 25 mm diameter shaft on the motor is made of a material having an allowable shear stress of allow = 75 MPa. If the motor is operating at its maximum power of 5 kw, determine the minimum allowable rotation of the shaft.
16 Torsion: angle of twist
17 Torsion: angle of twist BD x BD Shear strain: d dx Therefore: d dx
18 Torsion: angle of twist From before: dx d By Hook s law: T G G 1 ) ( ) ( 1 ), ( x x T G x Angle of twist: L dx x x T G x 0 ) ( ) ( 1 ) (
19 Torsion: angle of twist Constant torque and cross sectional area Angle of twist: ( x L) 1 G T L
20 Torsion: angle of twist Multiple torques x L AB i 1 G T L i L BC L CD
21 Torsion: example F The splined ends and gears attached to the A-36vsteel shaft are subjected to the torques shown. Determine the angle of twist of end B with respect to end A. The shaft has a diameter of 40 mm. x
22 Reading assignment Chapter 5 of textbook Review notes and text: ES2001, ES2501
23 Homework assignment As indicated on webpage of our course
WORCESTER POLYTECHNIC INSTITUTE
WORCESTER POLYTECHNIC INSTITUTE MECHANICAL ENGINEERING DEPARTMENT STRESS ANALYSIS ES-2502, C 2012 Lecture 03: Stress 17 January 2012 General information Instructor: Cosme Furlong HL-151 (508) 831-5126
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