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Mechanical Design of Machine Elements Complete after watching Module 16: Bending Stress Review

Module 16 Example: Bending Stress Review Below in figures 1 and 2 is rod OA, which is attached to another rod, AB. Assume that rod AB is strong enough, and stress analysis on rod AB is not part of this problem. Rod OA has a diameter of 4 cm. A force F = 1000 N is applied in the –x direction at the end of the rod OA, and a force P = 500 N is applied in the –y direction at point B. Assume that rod OA is made of a ductile metal. Find the bending stress at point O.

Figure 1: Isometric view of rod OA and rod AB.

Figure 2: Top-down view (X-Z plane) of rod OA and AB. Note that denotes force P at point B, which is going into the page.

Assumptions: Isotropic, homogenous, F is axial centric load, neglecting weight of rod OA and AB. Thoughts: - Force F creates an axial compressive stress in rod OA. . Force F does NOT contribute to the bending stress. - Force P creates a positive torsional shear stress in rod OA. We calculated this in the solution to worksheet 3. - Force P also creates a bending stress in rod OA. - Use of a free body diagram allows us to see that the force P also acts in the negative Y direction at point A. - The equation s = -(Mc)/I is what we use for bending stress - The moment M is created by load P and moment arm OA. - We are concerned about the bending stress in rod OA at point O. Point O is the distance ‘r’ away from the centric axis. Therefore we will use the radius of rod OA for ‘c’ and the diameter of rod OA for ‘I’ in s = -Mc/I. - The neutral axis is a plane, and lies along the X-Z plane at the center of the bar. - The bending stresses act in the x-direction, EVEN THOUGH the force ‘P’ acts in the ydirection. - The top of rod OA (at point O) is in tension, and has stresses acting in the positive xdirection. The bottom of rod OA is in compression, and has stresses acting in the negative xdirection.

Mechanical Design of Machine Elements Complete after watching Module 16: Bending Stress Review

Module 16 Example: Bending Stress Review Below in figures 1 and 2 is rod OA, which is attached to another rod, AB. Assume that rod AB is strong enough, and stress analysis on rod AB is not part of this problem. Rod OA has a diameter of 4 cm. A force F = 1000 N is applied in the –x direction at the end of the rod OA, and a force P = 500 N is applied in the –y direction at point B. Assume that rod OA is made of a ductile metal. Find the bending stress at point O. Figure 1: Isometric view of rod OA and rod AB.

Analysis: σ = - (Mc/I) M = P* lOA M = 500 N * 0.5 m = - 250 N*m c = dOA/2 I = ([πd4]/64) σ = -[M*(d/2)]/[([πd4])/64] σ

= - (32M)/(πd3) = -[32*(-250 N*m)]/[π*(0.04m)3] = 39.8 MPa (+ = tension)

Figure 4 : Neutral axis and bending stress distribution

Figure 2 : FBD of rod OA and rod AB.

Figure 3 : Bending Moment diagram of rod OA