1. Chapter 4: Stresses and Strains
I am never content until I have constructed a mechanical model of the subject I am studying. If I succeed in making one, I understand; otherwise I do not.
William Thomson (Lord Kelvin)
The failed Hyatt Regency Hotel walkway (Kansas City, 1981) that was directly attributable to changes in design that over-stressed structural members. (AP/Wide World Photos).

2. Centroid of Area The centroid is given by:
For complicated areas, break into
multiple areas:
Figure 4.1: Centroid of area. The centroid is at point C with coordinates (x,y). - -

3. Example 4.1
Figure 4.2: Rectangular hole within a rectangular section used in Example 4.1.

4. Area Moment of Inertia Area moment of inertia:
Polar moment of inertia:
Figure 4.3: Area with coordinates used in describing area moment of inertia.

5. Example 4.2
Figure 4.4: Circular cross section used in Example 4.2.

6. Parallel Axis Theorem Parallel axis theorem:
Figure 4.5: Coordinates and distance used in describing the parallel-axis theorem.

7. Examples 4.3 and 4.4
Figure 4.6: Triangular cross section with circular hole, used in Example 4.3.
Figure 4.7: Circular cross-sectional area relative to x’-y’ coordinates, used in Example 4.4.

8. Mass Element Radius of gyration: Section modulus:
Mass moment of inertia:
Figure 4.8: Mass element with three-dimensional coordinates.
Polar mass moment of inertia:

9. 2D Mass Element
Figure 4.9: Mass element in two-dimensional coordinates and distance from the two axes.

10. Section Properties
Table 4.1: Centroid, area moment of inertia, and area for common cross sections.

11. Mass and Mass Moment of Inertia
Table 4.2: Mass and mass moment of inertia of six solids.

12. Circular Bar and Normal Stress
Normal strain:
Deformation:
Spring rate:
Figure 4.10: Circular bar with tensile load applied.

13. Twisting of Bar and Shear
Shear stress:
Angle of twist:
Angular spring rate:
Figure 4.11: Twisting of member due to applied torque.
Power transfer:

14. Bending Bending stress distribution: Maximum bending stress:
Radius of curvature:
Figure 4.12: Bar made of elastic material to illustrate effect of applied bending moment. (a) Undeformed bar; (b) deformed bar.

15. Neutral Surface in Bending
Figure 4.13: Bending occurring in a cantilevered bar, showing neutral surface.

16. Deformation Elements in Bending
Figure 4.14: Undeformed and deformed elements in bending.

17. Bending Stress Variation
Figure 4.15: Profile view of bending stress variation.

18. Example 4.10
Figure 4.16: U-shaped cross section used in Example 4.10.

19. Curved Member in Bending
Figure 4.17: Curved member in bending. (a) Circumferential view; (b) cross-sectional view.

20. Cross Section of Curved Member
Figure 4.18: Rectangular cross section of curved member.

21. Transverse Shear Transverse shear stress:
Where Q is the first moment of area:
Figure 4.19: Development of transverse shear. (a) Boards not bonded together; (b) boards bonded together.

22. Transverse Shear Deformation
Figure 4.20: Cantilevered bar made of highly deformable material and marked with horizontal and vertical grid lines to show deformation due to transverse shear. (a) Undeformed; (b) deformed.

23. Stresses in Bending Segment
Figure 4.21: Three-dimensional and profile views of moments and stresses associated with shaded top segment of element that has been sectioned at y’ about neutral axis. (a) Three-dimensional view; (b) profile view.

24. Maximum Shear Stress
Table 4.3: Maximum shear stress for different beam cross sections.

25. Example 4.13
Figure 4.22: Shaft with loading considered in Example 4.13.
Figure 4.23: (a) Shear force; (b) normal force and (c) bending moment diagrams for the shaft in Fig

26. Example 4.13
Figure 4.24: Cross section of shaft at x = 100 mm, with identification of stress elements considered in Example 4.13.
Figure 4.25: Shear stress distributions. (a) Shear stress due to a vertical shear force; (b) Shear stress due to torsion.

27. Example 4.13 Results
Table 4.4: Stresses obtained in Example 4.13.

28. Coil Slitter
Figure 4.26: Design of shaft for coil slitting line. (a) Illustration of coil slitting line; (b) knife and shaft detail; (c) free-body diagram of simplified shaft for case study. Illustrations (a) and (b) are adapted from Tool and Manufacturing Engineers Handbook, Fourth Edition, Volume 2 Forming, (1984) Reprinted with permission of the Society of Manufacturing Engineers, ©1984.

29. Case Study Results
Figure 4.27: (a) Shear diagram and (b) moment diagram for idealized coil slitter shaft.
Figure 4.28: Mohr's circle at the location of maximum bending stress.