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).
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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).
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Example 4.1
Figure 4.2: Rectangular hole within a rectangular section used in Example 4.1.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Example 4.2
Figure 4.4: Circular cross section used in Example 4.2.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Parallel Axis Theorem
Parallel axis theorem:
Figure 4.5: Coordinates and distance
used in describing the parallel-axis
theorem.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Mass Element
Radius of gyration:
Section modulus:
Mass moment of inertia:
Figure 4.8: Mass element with threedimensional coordinates.
Polar mass moment of inertia:
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
2D Mass Element
Figure 4.9: Mass element in two-dimensional coordinates and distance from the
two axes.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Section
Properties
Table 4.1: Centroid, area moment
of inertia, and area for common
cross sections.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Mass and Mass
Moment of Inertia
Table 4.2: Mass and mass moment
of inertia of six solids.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Circular Bar and Normal Stress
Normal stress:
Normal strain:
Deformation:
Figure 4.10: Circular bar with tensile
load applied.
Spring rate:
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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:
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Neutral Surface in Bending
Figure 4.13: Bending occurring in a cantilevered bar, showing neutral
surface.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Deformation Elements in Bending
Figure 4.14: Undeformed and deformed elements in bending.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Bending Stress Variation
Figure 4.15: Profile view of bending stress variation.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Example 4.10
Figure 4.16: U-shaped cross section used in Example 4.10.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Curved Member in Bending
Figure 4.17: Curved member in bending. (a) Circumferential view; (b)
cross-sectional view.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Cross Section of Curved Member
Figure 4.18: Rectangular cross section of curved member.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Maximum Shear Stress
Table 4.3: Maximum shear stress
for different beam cross sections.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Example 4.13 Results
Table 4.4: Stresses obtained in Example 4.13.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press