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Unformatted text preview: Shigley’s Mechanical Engineering Design, 8 th Ed. Class Notes by: Dr. Ala Hijazi CH 4 Page 1 of 23 CH 4: Deflection and Stiffness Stress analysis is done to ensure that machine elements will not fail due to stress levels exceeding the allowable values. However, since we are dealing with deformable bodies (not rigid), deflections should be considered also where they are in many cases more limiting than stresses. Take for example shafts where excessive deflection will interfere with the function of the elements mounted on the shaft and might cause failure of the system, thus usually shafts are designed based on deflections rather than stresses. Spring Rates In most types of loading situations, the stress developed in the element (bar, shaft, beam, etc.) is linearly related to the loading. As long as the stress in the material remains within the linear elastic region, the stress is also linearly related to the deflection. Therefore, there is a linear relation between load and deflection and elements under loading behave similar to linear springs , and thus we can define the spring rate or spring constant for the element as: ܮܽ݀݅݊݃ ܦ݂݈݁݁ܿݐ݅݊ Tension, Compression and Torsion • For a bar with constant crosssection the deformation is found as: ߜ ൌ ܨ ܮ ܣ ܧ Thus, the spring constant for an axially loaded bar is: ݇ ൌ ܣ ܧ ܮ • For a round shaft subjected to torque, the angular deflection is found as: ߠ ൌ ܶܮ ܩܬ Axial, lateral, bending, twisting, etc. Axial, lateral, moment, torque, etc. ߠ in radians Shigley’s Mechanical Engineering Design, 8 th Ed. Class Notes by: Dr. Ala Hijazi CH 4 Page 2 of 23 Thus, the spring constant is: ܩ ܬ ܮ Deflection Due to Bending The deflection of beams is much larger than that of axially loaded elements, and thus the problem of bending is more critical in design than other types of deformation.  Shafts are treated as beams when analyzed for lateral deflection. • The beam governing equations are: Load intensity ா ூ ൌ ௗ ర ௬ ௗ௫ ర Shear force ாூ ൌ ௗ య ௬ ௗ௫ య Moment ெ ா ூ ൌ ௗ మ ௬ ௗ௫ మ Slope ߠ ൌ ௗ௬ ௗ௫ Deflection ݕ ൌ ݂ሺݔሻ • Knowing the load intensity function, we can integrate four times ( using the known boundary conditions to evaluate the integration constants ) to get the deflection. • However, in most cases, the expression of bending moment is easy to find ( using sections ) and thus we start from the moment governing equation and integrate to get the slope and deflection. Shigley’s Mechanical Engineering Design, 8 th Ed. Class Notes by: Dr. Ala Hijazi CH 4 Page 3 of 23 Example: The beam shown has constant crosssection and it is made from homogeneous isotropic material....
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This note was uploaded on 10/25/2010 for the course MECHINCAL 2010 taught by Professor علاءحجازي during the Spring '10 term at Hashemite University.
 Spring '10
 علاءحجازي

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