Page 86 - Instrumentation Reference Book 3E

easurernent of strain

B. E. NOLTINGK

F
4.1 Strain c=--tas
A
A particular case of length measurement is the when A is the area PQ x depth of block. In prac-
determination of strains, Le., the small changes tical situations, shear strain is often accompanied
in the dimensions of solid bodies as they are by bending, the magnitude of which is governed
subjected to forces. The emphasis on such mea- by Young’s modulus.
surements comes from the importance of know- There is some general concern with stress and
ing whether a structure is strong enough for its strain at all points in a solid body, but there is a
purpose or whether it may fail in use. particular interest in measuring strains on sur-
The interrelation between stress (the force per faces. It is only there that conventional strain
unit area) and strain (the fractional change in gauges can readily be used and wide experience
dimension) is a complex one: involving in general has been built up in interpreting results gained
three dimensions, particularly if the material con- with them. At a surface, the strain normal to it
cerned is not isotropic, Le., does not have the same can be calculated because the stress is zero (apart
properties ir, all directions. A simple stresshtrain from the exceptional case of a high fluid pressure
concept is of a uniform bar stretched lengthwise, being applied), but we still do not usually know
for which Young’s modulus of elasticity is defined the direction or magnitude of strains in the plane
as the ratio stress:strain. Le., the force per unit of of the surface so that for a complete analysis three
cross-sectional area divided by the fractional change strains must be measured i3 different directions.
in length The measurement of strain is also discussed in
Chapter 12.

The longitudinal extension is accompanied by 4.2 Bonded resistance strai
a transverse contraction. The ratio of the two gauges
fractions (transverse contraction)/(longitudinal
extension) is called Poisson’s ratio, denoted by In the early 194Qs, the bonded resistance strain
p., and is commonly about 0.3. While we have gauge was introduced and it has dominated the
talked of increases in length, called positive field of strain measurement ever since. Its princi-
strain, similar behavior occurs in compression, ple can be seen from Figure 4.2. A resistor R is
which is accompanied by a transverse expansion. bonded to an insulator I, which in turn is fixed to
Another concept is that of shear. Consider the the substrate S whose strain is to be measured.
block PQRS largely constrained in a holder, Fig- (The word “substrate” is not used universally with
ure 4.1. If this is subjected to a force F as shown, this meaning; it is adopted here for its conveni-
it will distort, PQ moving to P’Q’. The “shear ence and brevity.) When S is strained, the change
strain” is the ratio PP‘/PT, Le.. the angle PTP’ or in length is communicated to R if the bonding is
A$ (which equals angle QUQ) and the modulus adequate; it can be shown that the strain will be
ofrigidity is defined as (shear stress)/(shear strain) transmitted accurately even through a mechanic-
or ally compliant insulator provided there is suffi-
cient overlap, Le.. if I is larger than R by severai

S
Figure 4.1 Shear strain. Figure 4.2 Principle of resistance strain gauge