experiment ii the resistance strain gauge objectives the objectives of
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EXPERIMENT II
The Resistance Strain Gauge
OBJECTIVES
The objectives of this experiment are (a) to become familiar with the use of the resistance strain
gauge, (b) to compare data obtained by strain gauge measurements with those from extensometer
measurements, and (c) to use statistical methods as part of the data analysis.
THEORY
Strain measurements using the resistance strain gauge are based on the change in electrical
resistance of a thin wire when this wire deforms. The resistance R of a wire depends on the
length (L) and area (4) of the wire according to the following equation:
R =
P
L
(II1)
where p is the resistivity of the wire. Since the volume of the wire is V = LA, Eq. (II1) can also
be written as:
R
=
ve
V
(II2)
If the thin wire is bonded securely to an object which is being strained, this strain will change the
length of the wire from L to L+AL. According to Eq. (II2), this change in length will cause a
change in resistance. Assuming the volume, V, does not change during this process (it actually
changes slightly), and that the resistivity of the material remains constant, it follows from Eq. (II
2) that
and
log R
=
2 log L log
log
dR
R
dL
=
2.
(II3)
L
For finite, but small, changes Eq. (II3) becomes
AR
R
=
ΔΙ
F
(II4)
L
II1 and
1 AR
ε =
FR
(II5)
where F≈2 is the gauge factor, since the strain is defined by &= AL/L. In other words, the strain
of the gauge is directly proportional to the fractional change in resistance. For actual gauges, F
is usually slightly greater than the value of 2, and is supplied by the strain gauge manufacturer.
In this experiment, the gauge has a factor of 2.140. Additional information on strain gage
selection can be found on the datasheets provided by strain gauge manufacturers. Refer, for
example, to http://www.vishay.com/docs/11055/tn505.pdf.
From Experiment I, it is realized that strains of practical interest are very small. Accordingly, to
employ the changes in resistance, we use a resistance bridge (a Wheatstone bridge) where the
gauge resistance is compared to a known, calibrated resistance.
The resistance of the strain gauge is also affected by changes in temperature, and by leakage of
the current through the bonding material to the object. To minimize these effects, a compensating
gauge is used which is bonded to an identical but unstrained material at the same temperature.
This setup reads the strain directly when the gauge factor is 2.00. Whenever F differs from this
value it is necessary to correct the strains, as follows:
Ес
2 &r
F
(II6)
where
&c=corrected strain
ε = strain read by the data acquisition device.
Important
Please note that, in this experiment, the data acquisition software has been programmed to do the
above correction internally.
II2 CURVE FITTING BY METHOD OF LEAST SQUARES
The following is a brief overview of the theory of the method of least squares. Suppose that n
number of data points (xi, y;) ... (xn, y), concerning two variables x and y, have been collected
at the end of an experiment and are identified on an xy graph. It is desired to fit a straight line
ŷ = ax + b
(II7)
II5 through the given points so that the sum of the squares of the distances of those points from the
straight line is minimum, where the distance is measured in the y (vertical) direction. The
absolute error associated with a pair of data points is
e; = y; — ŷ;=y; − (ax; +b)

(II8)
Therefore, the equation for the sum of the squares of the error between the straight line and the
data points becomes
n
n
Q = Σ e² = Σ[y; − ax; —b)]²
i=1
i=1
(II9)
In order to minimize the total error, the derivatives of Q with respect to a and b are set equal to
zero.
до
да
=
n
2 Σ
i=1
‚y, – aɖx,
nb
= 0
i=1
n
n
ΣΟ
2xy  bx;  a
bĹx, − aɖ(x,)³] = 0
до
ab
i=1
By definition, the mean values of the data points are given by
i=1
n
n
Σν
i=1
and
y
n
Therefore, solving for a and b, Equations II10 and II11 simultaneously yield:
n
i=1
a =
n
xy; nx y
Σx²nx²
i=1
223
b =
i=1
n
i=1
Στ
Σx²  nx²
i=1
2
The standard error of estimate of the computed value of ŷ is defined as:
II6
(II10)
(II11)
(II12)
(II13) n
Σ (y; – ŷ₁ )²
SE =
i=1
n2
(II14)
REPORT REQUIREMENTS
1. Graph the least squares fit for points representing stress vs. strain using the strain gauge
readings.
2. Graph the least squares fit for points representing stress vs. strain from the direct
extensometer readings.
3. Compare the modulus of elasticity from each of the above graphs with the published value
for aluminum by calculating the Percent Error (using the definition below).
Percent Error: Applied when comparing an experimental quantity, E, with a theoretical
quantity, T, which is considered the "correct" value. The percent error is the absolute value of the
difference divided by the "correct" value times 100.
% Error:
TE
T
•100
REFERENCES
[1] Holman, J.P, Experimental Methods for Engineers, McGrawHill.
[2] Beer, Johnston, DeWolf, and Mazurek. Mechanics of Materials, McGrawHill.
II7