E1C11 09/14/2010
13:14:3 Page 488
application (5). In practice, the applicability of the single-axis strain gauge is extremely limited, and
improper use can result in large errors in the measured stress.
Strain gauge rosettes are commercially available in a variety of configurations, including gauges
placed at angles of 0, 45, 90 degrees and 60, 60, 60 degrees. Consider the rectangular rosettes with
angles of 0, 45, 90 degrees as shown in Figure 11.17. As we have emphasized, measuring surface
strain allows calculation of the principal stresses. Assume that the measured strains from the stacked
gauge in Figure 11.17b are e 1 (from the 0-degree gauge, oriented with the x-axis) e 2 (from the
45-degree gauge), and e 3 (from the 90-degree gauge, oriented with the y-axis). From these measured strains, the principal stresses, s max and s min , and the maximum shear stress, t max , may be
calculated as
s max ¼
E m
2
e 1 þ e 3
1 À y p
þ
1
1 þ y p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e 1 À e 3
ð
Þ
2 þ 2e 2 À e 1 þ e 3
ð
Þ
½
2
q
!
s min ¼
E m
2
e 1 þ e 3
1 À y p
À
1
1 þ y p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e 1 À e 3
ð
Þ
2 þ 2e 2 À e 1 þ e 3
ð
Þ
½
2
q
!
t max ¼
E m
2 1 þ y p
À
Á
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e 1 À e 3
ð
Þ
2 þ 2e 2 À e 1 þ e 3
ð
Þ
½
2
q
ð11:39Þ
The angle between the x-axis and the maximum principal stress is given by
f ¼
1
2
tan
À1 2e 2 À e 1 þ e 3
ð
Þ
e 1 À e 3
ð11:40Þ
Figure 11.16 Biaxial strain gauge rosettes. (a) Single-plane
type. (b) Stacked type. (Courtesy of Micro-Measurements
Division, Measurements Group, Inc., Raleigh, NC.)
488 Chapter 11 Strain Measurement
13:14:3 Page 488
application (5). In practice, the applicability of the single-axis strain gauge is extremely limited, and
improper use can result in large errors in the measured stress.
Strain gauge rosettes are commercially available in a variety of configurations, including gauges
placed at angles of 0, 45, 90 degrees and 60, 60, 60 degrees. Consider the rectangular rosettes with
angles of 0, 45, 90 degrees as shown in Figure 11.17. As we have emphasized, measuring surface
strain allows calculation of the principal stresses. Assume that the measured strains from the stacked
gauge in Figure 11.17b are e 1 (from the 0-degree gauge, oriented with the x-axis) e 2 (from the
45-degree gauge), and e 3 (from the 90-degree gauge, oriented with the y-axis). From these measured strains, the principal stresses, s max and s min , and the maximum shear stress, t max , may be
calculated as
s max ¼
E m
2
e 1 þ e 3
1 À y p
þ
1
1 þ y p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e 1 À e 3
ð
Þ
2 þ 2e 2 À e 1 þ e 3
ð
Þ
½
2
q
!
s min ¼
E m
2
e 1 þ e 3
1 À y p
À
1
1 þ y p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e 1 À e 3
ð
Þ
2 þ 2e 2 À e 1 þ e 3
ð
Þ
½
2
q
!
t max ¼
E m
2 1 þ y p
À
Á
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e 1 À e 3
ð
Þ
2 þ 2e 2 À e 1 þ e 3
ð
Þ
½
2
q
ð11:39Þ
The angle between the x-axis and the maximum principal stress is given by
f ¼
1
2
tan
À1 2e 2 À e 1 þ e 3
ð
Þ
e 1 À e 3
ð11:40Þ
Figure 11.16 Biaxial strain gauge rosettes. (a) Single-plane
type. (b) Stacked type. (Courtesy of Micro-Measurements
Division, Measurements Group, Inc., Raleigh, NC.)
488 Chapter 11 Strain Measurement