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

Also, Equation (13.17) becomes, simply,
γ
xy
1
= τxy
(13.25)
G
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gENERALIzEd HOOkE’S LAw
FOR ISOTROPIC MATERIALS
Then, solving Equation (13.24) for the stresses in terms of the strains gives the following result:
σ
σ
E
= ( ε
1 −
2
ν
+ νε )
E
= ( ε
2
1 − ν
+ νε )
x x y
y y x
(13.26)
Equation (13.26) can be used to calculate normal stresses from measured or computed
normal strains.
Note that the out-of-plane normal strain ε z is generally not equal to zero for the plane
stress condition. An expression for ε z in terms of ε x and ε y was stated in Equation (13.15).
This equation can be derived by substituting Equation (13.26) into the expression
to give
ν =−
E ( σ + σ )
ε z x y
ν ν E
ε z =− ( σ x + σ y)
= − [( εx + νεy) + ( εy + νε x )]
2
E
E1
− ν
ν
=−
[(1 + νε ) x + (1 + νε ) y]
(1 − ν)(1 + ν) ν
=− ( x y)
1 ν ε + ε −
(13.27)
ExAmpLE 13.7
On the free surface of an aluminum [E = 10,000 ksi; ν = 0.33] component, three strain
gages arranged as shown record the following strains: ε a = −420 µε, ε b = 380 µε, and
ε c = 240 µε.
Determine the normal stress that acts along the axis of gage b (i.e., at an angle of
θ = 45° with respect to the positive x axis).
Plan the Solution
At first glance, one might be tempted to use the measured strain in gage b and the elastic
modulus E to compute the normal strain acting in the specified direction. However, that approach
is not correct because a state of uniaxial stress does not exist. In other words, the
normal stress acting in the 45° direction is not the only stress acting in the material. To solve
this problem, first reduce the strain rosette data to obtain ε x , ε y , and γ xy . The stresses σ x , σ y ,
and τ xy can then be calculated from Equations (13.26) and (13.25). Finally, the normal stress
in the specified direction can be calculated from the stress transformation equation.
c
y
45°
45°
b
a
x
SolutioN
From the geometry of the rosette, gage a measures the strain in the x direction and gage c
measures the strain in the y direction. Therefore, ε x = −420 µε and ε y = 240 µε. To compute
the shear strain γ xy , write a strain transformation equation for gage b:
2 2
380 = ε cos (45 ° ) + ε sin (45 ° ) + γ sin(45 ° )cos(45 ° )
x y xy