Object
Use strain gauges to measure the strain in a loaded steel cantilever.
Verify Hooke's Law.
Verify the calculation for the strain in a loaded cantilever.
Introduction
The electrical resistance (R) of a metal wire is
given by (where r
is the resistivity), so the resistance is proportional to the
length (L) and inversely proportional to the area (A). As the
wire stretches it becomes longer and thinner (because the volume
of the wire stays approximately the same). Hence the resistance
of the wire is increased by stretching. When a wire is compressed
it becomes shorter and fatter, this reduces the resistance. The
material structure changes slightly during stretching and compression,
this produce small resistivity changes.
Strain Gauges are thin wires that can be glued to a metal structure. When the structure flexes under a load the resistance of the strain gauges changes and this can be used to measure the strain in the structure. In this way, the strain in a structure (e.g. an oil rig or an aircraft wing) can be measured to verify the design calculations.
Theory
The change in resistance DR in a strain gauge of resistance R is very nearly proportional to the applied strain. Hence:
K is a constant known as the gauge factor and is
the relative strain . The gauges used
in this experiment have K = 2.10 ± 0.02.
Figure 1: Loaded Cantilever Beam.
The gauges are a distance D from the load (see figure 1), a load of mass m and weight mg is suspended from the cantilever beam (g is the acceleration due to gravity). The beam has thickness t and width w and is made from stainless steel with a Young's Modulus E. The calculated strain due to the suspended mass is:
Therefore the relative change in the resistance of the strain gauge is given by:
The resistance changes in the strain gauges are very small, therefore the gauges are connected in a Wheatstone Bridge Circuit (see figure 2). The gauge on top of the beam is in tension, the gauge underneath the beam is in compression, hence strain causes equal and opposite resistance changes in the gauges. By using two gauges the effects of temperature variations on the gauge resistances are cancelled out.
The left hand end of the bridge circuit is at zero volts (see figure 1), the circuit is powered by the bridge excitation voltage V_{EX} applied to the right hand end of the bridge (see figure 1).
If the strain increases the resistance of Gauge One from R to R + DR then the resistance of Gauge Two is decreased from R to R - DR. Hence the voltage V_{G} (see figure 2) is given by:
To balance the Wheatstone Bridge the Zero Adjust resistor is adjusted to produce a voltage of . Therefore the output voltage V_{o} of the Wheatstone bridge is given by:
Substituting then:
Figure 2: Strain Gauge Wheatstone Bridge Circuit.
Apparatus
Cantilever Beam apparatus, Bridge Power Supply, high sensitivity voltmeter, Wheatstone Bridge, digital voltmeter, 1 kg set of 100g masses (individually marked with accurate masses).
Preliminary Experiment
Setting up the Wheatstone Bridge Circuit
Experiment
Symbol | Description | Value | Units |
K | Gauge Factor | 2.10 ± 0.02 | dimensionless |
m | suspended mass | ± 0.01 per mass | grams (g) |
g | acceleration due to gravity | 9.816 ± 0.001 | metres per second squared (ms^{-2}) |
D | load to gauge distance | 0.504 ± 0.003 | metres (m) |
V_{EX} | bridge excitation voltage | estimated error | volts (V) |
E | Young's modulus | 195 ± 5 | Giga Pascals (GPa) |
w | width (horizontal) of cantilever | 2.53 ± 0.01 | centimetres (cm) |
t | thickness (vertical) of cantilever | 3.55 ± 0.05 | millimetres (mm) |
© Mark Davison, 1997, give feedback or ask questions about this experiment.
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