CH 3 - CH 3: Load and Stress Analysis Machine elements...

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Shigley’s Mechanical Engineering Design, 8 th Ed. Class Notes by: Dr. Ala Hijazi CH 3 Page 1 of 30 CH 3: Load and Stress Analysis Machine elements carry different types of loads ( concentrated, distributed, axial, lateral, moments, torsion, etc. ) according to the function and configuration of each element. These loads cause stresses of different types and magnitudes in different locations in the element. When designing machine elements it is important to locate the critical locations ( or sections ) and to evaluate the stress at the critical sections to ensure the safety and functionality of the machine element. Equilibrium and Free-Body Diagrams Equilibrium of a body requires both a balance of forces (to prevent translation) and balance of moments (to prevent rotations). ෍ܨ ൌ Ͳ ෍ܯ ൌ Ͳ A free body diagram ( FBD ) is a sketch of an element or group of connected elements that shows all the forces acting on it ( applied loads, gravity forces, and reactions ) See Example 3-1 from text
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Shigley’s Mechanical Engineering Design, 8 th Ed. Class Notes by: Dr. Ala Hijazi CH 3 Page 2 of 30 Shear and Moment in Beams Shear and moment diagrams are important in locating the critical sections in a beam ( sections with maximum shear or moment ) such that stresses are evaluated at these sections. The sign convection for shear force and bending moment is: Shear force and bending moment are related by the equation ܸ ൌ ௗெ ௗ௫ ( Shear is the slope of the moment diagram ) When a distributed load ݍሺݔሻ is applied to the beam ݍ ൌ ݀ݒ ݀ݔ ݀ ܯ ݀ݔ Integrating the relations we get: න ݍ ݀ݔ ൌ න ܸ݀ ൌ ܸ െ ܸ න ܸ ݀ݔ ൌ න ݀ܯ ൌ ܯ െ ܯ To draw shear and moment diagrams: - Draw the FBD & find reactions using equilibrium. - Make sections and draw their FBD and find ܸ Ƭ ܯ . The change in shear force between sections is equal to the area under the loading diagram The change in moment between 1 & 2 is equal to the area under the shear diagram
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th Ed. Class Notes by: Dr. Ala Hijazi CH 3 Page 3 of 30 Example: Draw the shear & moment diagrams for the beam shown. Solution : ૙ ൏ ݔ ൏ ʹ b ܸ ൌ െͲǤͷ ݇ܰ b ܯ ൌ െͲǤͷݔ ݇ܰǤ݉ ૛ ൏ ݔ ൏ ͵ b ܸ ൌ െͲǤͷ ݇ܰ ܯ െ ͵ ൅ ͲǤͷݔ ൌ Ͳ b ܯ ൌ ͵ െ ͲǤͷݔ ݇ܰǤ݉ ૜ ൏ ݔ ൏ Ͷ െͲǤͷ ൅ ʹ െ ܸ ൌ Ͳ b ܸ ൌ ͳǤͷ ݇ܰ ܯ െ ʹሺݔ െ ͵ሻ െ ͵ ൅ ͲǤͷݔ ൌ Ͳ b ܯ ൌ ͳǤͷݔ െ ͵ ݇ܰǤ݉ ૝ ൏ ݔ ൏ ͸ െͲǤͷ ൅ ʹ െ ͵ሺݔ െ Ͷሻ െ ܸ ൌ Ͳ b ܸ ൌ െ͵ݔ ൅ ͳ͵Ǥͷ ݇ܰ ܯ ൅ ͵ሺݔ െ Ͷሻቈ ሺݔ െ Ͷሻ ʹ ቉ െ ʹሺݔ െ ͵ሻ െ͵ ൅ ͲǤͷݔ ൌ Ͳ b ܯ ൌ െͳǤͷݔ ൅ ͳ͵Ǥͷݔ െ ʹ͹ ݇ܰǤ݉ ෍ܯ ൌ Ͳ ෍ܨ ൌ Ͳ ܴ ൅ ʹ െ ͸ ൅ ͶǤͷ ൌ Ͳ െ͵ ൅ ʹሺ͵ሻ െ ͸ሺͷሻ ൅ ܴ
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CH 3 - CH 3: Load and Stress Analysis Machine elements...

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