Strain Gauges and the Wheatstone Bridge

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 VEX 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 VG (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 Vo 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

1. With the cantilever beam unloaded, measure and note the resistance of one of the gauges using a Digital Multimeter (in resistance mode). What is the change in gauge resistance when a 1 kg load is suspended from the beam?

Setting up the Wheatstone Bridge Circuit

1. Assemble the Wheatstone Bridge Circuit as shown in figure 2.
2. Connect the high sensitivity voltmeter to measure output voltage.
3. The Bridge Power Supply and the digital multimeter (in voltage mode) used to measure the Bridge Excitation Voltage are connected using separate wires. This eliminates errors due to the resistance in the wires and connections.
4. Adjust the Zero Adjust resistor to obtain a zero output voltage when the cantilever beam is unloaded.

Experiment

1. Measure the bridge output voltage with beam loadings from 0.00 to 1.00 kg in 0.10 kg steps.
2. Plot a graph of output voltage (in Volts) on the vertical axis versus beam loading (in kg) on the horizontal axis.
3. Hooke's Law states that for elastic behaviour that the strain is proportional to the load applied, does your graph verify Hooke's Law?
4. If the design of the apparatus is correct, the output voltage of the strain gauge circuit should be . Use this formula to calculate the output voltage you expect for one of the loadings you have measured.
5. Calculate the maximum experimental error in the expected value of Vo using the formula .
6. Does the output voltage you expect agree with output voltage you have measured within their respective experimental errors? What does this tell you?

 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) VEX 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)