Chapter 5: Deformation

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Chapter 5: Deformation

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Testing of 787 Dreamliner wings. (Courtesy of Boeing Corp.)

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Moment-Curvature Relations

In terms of ordered derivatives:

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Design Procedure 5.1: Deflection by Singularity Functions

1. Draw a free-body diagram showing the forces acting on the system.

2. Use force and moment equilibria to establish reaction forces acting on the system.

3. Obtain an expression for the load intensity function for all the loads acting on the system while making use of Table 2.2.

4. Integrate the negative load intensity function to give the shear force and then integrate the negative shear force to give the moment.

5. Make use of Eq. (5.9) to describe the deflection at any location.6. Plot the following as a function of x:

a. Shearb. Momentc. Sloped. Deflection

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Beams

Figure 5.1: Cantilevered beam with concentrated force applied at free end. Figure 5.2: Free-body diagram of

force anywhere between simply supported ends. (a) Complete beam; (b) portion of beam.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Example 5.3

Figure 5.3: Cantilevered beam with unit step distribution over part of beam. (a) Loads and deflection acting on cantilevered beam; (b) free-body diagram of forces and moments acting on entire beam; (c) free-body diagram of forces and moments acting on portion of beam.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Example 5.4

Figure 5.4: Pinned-fixed beam with concentrated force acting anywhere along beam. (a) Sketch of assembly; (b) free-body diagram of entire beam; (c) free-body diagram of part of beam.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Beam Deflection

Table 5.1: Deflection for common cantilever and simply-supported beam conditions. See also Appendix D.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Example 5.5

Figure 5.5: Beam fixed at one end and free at other with moment applied to free end and concentrated force at any distance from free end. (a) Complete assembly; (b) free-body diagram showing effect of concentrated force; (c) free-body diagram showing effect of moment.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Stress Elements

Figure 5.6: Element subjected to normal stress.

Figure 5.7: Element subjected to shear stress.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Strain Energy

Table 5.2: Strain energy for four types of loading.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Example 5.8

Figure 5.8: Cantilevered beam with concentrated force acting at a distance b from free end. (a) Coordinate system and significant points shown; (b) fictitious force, Q, shown along with concentrated force, P.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Example 5.9

Figure 5.9: System arrangement. (a) Entire assembly; (b) free-body diagram of forces acting at point A.

Fundamentals of Machine Elements, 3rd ed.Schmid, Hamrock and Jacobson © 2014 CRC Press

Example 5.10

Figure 5.10: Cantilevered beam with 90° bend acted upon by horizontal force, P, at free end.