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

Equations (13.9), (13.10), and (13.11) or the corresponding Mohr’s circle construction can
be used to determine the in-plane principal strains, their orientations, and the maximum
in-plane shear strain at a point.
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STRAIN MEASuREMENT ANd
STRAIN ROSETTES
Strains in the out-of-plane Direction
Rosettes are bonded to the surface of an object, and stresses in the out-of-plane direction
on the free surface of an object are always zero. Consequently, a state of plane stress exists
at the rosette. Whereas strains in the out-of-plane direction are zero for the plane strain
condition, out-of-plane strains are not zero for plane stress.
The principal strain e z = e p3 can be determined from the measured in-plane data with
the equation
=− ν
( )
1 − ν e + e
(13.15)
e z x y
where ν = Poisson’s ratio. The derivation of this equation will be presented in the next section,
in the discussion of the generalized Hooke’s law. The out-of-plane principal strain is
important, since the absolute maximum shear strain at a point may be (e p1 − e p2 ), (e p1 − e p3 ),
or (e p3 − e p2 ), depending on the relative magnitudes and signs of the principal strains at the
point. (See Section 13.4.)
ExAmpLE 13.4
A strain rosette consisting of three strain gages oriented as illustrated was mounted on the
free surface of a steel machine component (ν = 0.30). Under load, the following strains
were measured:
e =−600 me e =−900 me e = 700 me
a b c
Determine the principal strains and the maximum shear strain at the point shown. Show the
principal strain deformations and the maximum in-plane shear strain distortion in a sketch.
b
c
90°
y
135°
135°
a
x
Plan the Solution
To compute the principal strains and the maximum in-plane shear strain, the strains e x , e y ,
and γ xy must be determined. These normal and shear strains can be obtained from the
rosette data by writing a strain transformation equation for each gage and then solving the
three equations simultaneously. Since it is aligned with the x axis, gage a directly measures
the normal strain ε x , so the reduction of the strain gage data will actually involve
solving only two equations simultaneously, for ε y and γ xy .
SolutioN
The angles θ a , θ b , and θ c must be determined for the three gages. Although it is not an
absolute requirement, strain rosette problems such as this one are easier to solve if all
angles θ are measured counterclockwise from the reference x axis. For the rosette configuration
used in this problem, the three angles are θ a = 0°, θ b = 135°, and θ c = 225°.
Using these angles, write a strain transformation equation for each gage, where ε n is the
experimentally measured strain value:
Equation for gage a:
2 2
− 600 = ε cos (0) ° + ε sin (0) ° + γ sin(0)cos(0) ° °
x y xy
(a)