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).

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). - -

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

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.

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

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

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.

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:

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

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

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

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

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:

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.

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

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

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

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

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

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

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.

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.

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.

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

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. 4.22.

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.

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

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.

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.