Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Equation for gage b:
2 2
− 900 = e cos (135 ° ) + e sin (135 ° ) + γ sin(135 ° )cos(135 ° )
x y xy
Equation for gage c:
2 2
700 = ε cos (225 ° ) + ε sin (225 ° ) + γ sin(225 ° )cos(225 ° )
x y xy
(b)
(c)
Since sin(0°) = 0, Equation (a) reduces to ε x = −600 µε. Substitute this result into Equations
(b) and (c), and collect constant terms on the left-hand side of the equations:
− 600 = 0.5ε
− 0.5γ
1, 000 = 0.5ε
+ 0.5γ
Generally, the gage orientations used in common rosette patterns produce a pair of equations
similar in form to these two equations, making them especially easy to solve simultaneously.
To obtain ε y , the two equations are added together to give ε y = 400 µε. Subtracting
the two equations results in γ xy = 1,600 µrad. Therefore, the state of strain that
exists at the point on the steel machine component can be summarized as ε x = −600 µε,
ε y = 400 µε, and γ xy = 1,600 µrad. These strains will be used to determine the principal
strains and the maximum in-plane shear strain.
From Equation (13.10), the principal strains can be calculated as
y
y
xy
xy
ε
p1, p2
εx + εy ⎛εx − εy⎞
γ xy
= ± ⎜ ⎟ + ⎛ 2 ⎝ 2 ⎠ ⎝ ⎜
⎞
⎟
2 ⎠
600 400
= − + ±
2
=− 100 ± 943
2 2
2 2
⎛−600 − 400 ⎞ 1,600
⎜ ⎟ + ⎛ ⎝ 2 ⎠ ⎝ ⎜ ⎞
⎟
2 ⎠
= 843 µε− , 1,043 µε Ans.
and from Equation (13.11), the maximum in-plane shear strain is
2 2
γ max ⎛ εx − εy⎞
γ xy
=±
2 ⎝
⎜
2 ⎠
⎟ + ⎛ ⎝ ⎜ ⎞
2 ⎠
⎟
2 2
⎛ −600 − 400⎞
1,600
=±
⎝
⎜
2 ⎠
⎟ + ⎛ ⎝ ⎜ ⎞
2 ⎠
⎟
= 943.4 µ rad
∴ γ = 1,887 µ rad
Ans.
The in-plane principal directions can be determined from Equation (13.9):
tan2θ
p
x
xy
y
max
1,600 1,600
= = Note: εx
− εy
< 0
− 600 − 400 − 1, 000
∴ 2θ
= − 58.0° and thus θ =− 29.0°
p
=
ε
γ
− ε
Since ε x − ε y < 0, the angle θ p is the angle between the x direction and the ε p2 direction.
The strain rosette is bonded to the surface of the steel machine component;
therefore, the condition in this example is a plane stress condition. Accordingly, the
p
558