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

ExAmpLE 13.9
On the free surface of a copper alloy [E = 115 GPa; ν = 0.307] component, three strain
gages arranged as shown record the following strains at a point:
y
e = 350 me e = 990 me e = 900 me
a b c
(a) Determine the strain components e x , e y , and γ xy at the point.
(b) Determine the principal strains and the maximum in-plane shear strain at the point.
(c) Using the results from part (b), determine the principal stresses and the maximum
in-plane shear stress. Show these stresses in an appropriate sketch that indicates the
orientation of the principal planes and the planes of maximum in-plane shear stress.
(d) Determine the magnitude of the absolute maximum shear stress at the point.
a
45°
45°
b
c
x
Plan the Solution
To solve this problem, first reduce the strain rosette data to obtain ε x , ε y , and γ xy . Then, use
Equations (13.9), (13.10), and (13.11) to determine the principal strains, the maximum
in-plane shear strain, and the orientation of these strains. The principal stresses can be
calculated from the principal strains with Equation (13.26), and the maximum in-plane
shear stress can be computed from Equation (13.25).
SolutioN
(a) Strain Components ε x , ε y , and γ xy
To reduce the strain rosette data, the angles θ a , θ b , and θ c must be determined for the three
gages. For the rosette configuration used in this problem, the three angles are θ a = 45°,
θ b = 90°, and θ c = 135°. (Alternatively, the angles θ a = 225°, θ b = 270°, and θ c = 315°
could be used.) Using these angles, write a strain transformation equation for each gage,
where the strain ε n is the experimentally measured value:
Equation for gage a:
2 2
350 = ε cos (45 ° ) + ε sin (45 ° ) + γ sin(45 ° )cos(45 ° )
x y xy
(a)
Equation for gage b:
2 2
990 = ε cos (90 ° ) + ε sin (90 ° ) + γ sin(90 ° )cos(90 ° )
x y xy
(b)
Equation for gage c:
2 2
900 = ε cos (135 ° ) + ε sin (135 ° ) + γ sin(135 ° )cos(135 ° )
x y xy
(c)
Since cos(90°) = 0, Equation (b) reduces to ε y = 990 µε. Substitute this result into Equations
(a) and (c), and collect the constant terms on the left-hand side of the equations:
− 145 = 0.5ε
+ 0.5γ
405 = 0.5ε
− 0.5γ
These two equations are added together to give ε x = 260 µε. Subtracting the two equations
gives γ xy = −550 µrad. Therefore, the state of strain that exists at the point on the copper
alloy component can be summarized as ε x = 260 µε, ε y = 990 µε, and γ xy = −550 µrad.
These strains will be used to determine the principal strains and the maximum in-plane
shear strain.
Ans.
x
x
xy
xy
571