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

556
STRAIN TRANSFORMATIONS
Plastic
backing
Metal-foil
sensing
grid
FIGURE 13.7
Alignment
marks
Solder
tabs
The rosette shown in Figure 13.8
is called a rectangular rosette
because the angle between gages
is 45°. The rectangular rosette is
the most common rosette pattern.
In this equation, ∆R is the resistance change and ∆L is the change in length of the strain
gage. The gage factor is constant for the small range of resistance change normally encountered,
and most typical gages have a gage factor of about 2. Strain gages are very accurate,
relatively inexpensive, and reasonably durable if they are properly protected from chemical
attack, environmental conditions (such as temperature and humidity), and physical damage.
Strain gages can measure normal static and dynamic strains as small as 1 × 10 −6 .
The photoetching process used to create the metal-foil grids is very versatile, enabling
a wide variety of gage sizes and grid shapes to be produced. A typical single strain gage is
shown in Figure 13.7. Since the foil itself is fragile and easily torn, the grid is bonded to a
thin plastic backing film, which provides both strength and electrical insulation between
the strain gage and the object being tested. For general-purpose strain gage applications, a
polyimide plastic that is tough and flexible is used for the backing. Alignment markings are
added to the backing to facilitate proper installation. Lead wires are attached to the solder
tabs of the gage so that the change in resistance can be monitored with a suitable instrumentation
system.
The objective of experimental stress analysis is to determine the state of stress at a
specific point in the object being tested. In other words, the investigator ultimately wants to
determine σ x , σ y , and τ xy at a point. To accomplish this task, strain gages are used to determine
e x , e y , and γ xy , and then stress–strain relationships are used to compute the corresponding
stresses. However, strain gages can measure normal strains in only one direction.
Therefore, the question becomes “How can one determine three quantities (ε x , ε y , and γ xy )
by using a component that measures normal strain ε in only a single direction?”
The strain transformation equation for the normal strain ε n at an arbitrary direction θ
was derived as Equation (13.3) in Section 13.3:
2 2
ε = ε cos θ + ε sin θ + γ sinθcosθ
n n y xy
Suppose that ε n could be measured by a strain gage oriented at a known angle θ. Three
unknown variables—ε x , ε y , and γ xy —remain in Equation (13.3). To solve for these three
unknowns, three equations in terms of ε x , ε y , and γ xy are required. Those equations can be
obtained by using three strain gages in combination, with each gage measuring the strain in
a different direction. This combination of strain gages is called a strain rosette.
FIGURE 13.8 Typical strain
rosette.
Strain Rosettes
A typical strain rosette is shown in Figure 13.8. The gage is configured so that the angles
between each of the three gages are known. When the rosette is bonded to the object being
tested, one of the three gages is aligned with a reference axis on the object—for example,
along the longitudinal axis of a beam or a shaft. During the experimental test, strains are
measured from each of the three gages. A strain transformation equation can be written for
each of those gages in the notation indicated in Figure 13.9:
Gage
b
y
θ b
Gage
a
2 2
ε = ε cos θ + ε sin θ + γ sinθ cosθ
a x a y a xy a a
2 2
ε = ε cos θ + ε sin θ + γ sinθ cosθ
b x b y b xy b b
2 2
ε = ε cos θ + ε sin θ + γ sinθ cosθ
c x c y c xy c c
(13.14)
Gage
c
θ c
FIGURE 13.9
θ a
x
In this book, the angle used to identify the orientation of each rosette gage will always be
measured counterclockwise from the reference x axis.
The three strain transformation equations in Equation (13.14) can be solved simultaneously
to yield the values of ε x , ε y , and γ xy . Once ε x , ε y , and γ xy have been determined,