650:349 Mechanical Measurements Group 4B January 18, 2006

Strain Gage Measurements of Constant Stress and Cantilever Beams

Your Name Here[1], Norwood Fisher, Lisa Grant, Walter Kibby, Kendall Jones, Chris Dowd, Angelo Moore

Department of Mechanical

and Aerospace Engineering

Rutgers University, Piscataway, New Jersey 08854

Axial and transverse strain is measured in both constant stress standard cantilever beams using single strain gages and strain gages rosettes for several different metals. The Young’s moduli of steel, brass, and aluminum are measured to be 299 GPa, 44 GPa, and 22 GPa respectively. The Poisson’s ratio of steel, brass, and aluminum are measured to be 0.54, 0.31, and 0.22 respectively. Strain measurements using a standard Wheatstone Bridge/Digital Volt Meter arrangement are compared to those made using a Strain Gage Indicator. Finally, multi-axis strain measurements are used to determine the principal stresses in a cantilever beam.

INTRODUCTION

Strain measurements are commonly used to determine the local deformation of materials to applied loads. These are frequently used in mechanical engineering and materials science applications. The strain gage is a small device which changes its resistance when stretched in its primary axis. By incorporating the strain gage element into a Wheatstone bridge (a circuit which can be made sensitive to small changes in resistance), a varying voltage can be recorded which is proportional to the strain.

This laboratory uses two different kinds of beams. The first is a constant stress beam, with a width which varies linearly along its length. Due to this variation, the axial stress is constant along the length of the beam, which is not the case in a traditional cantilever beam. An example constant stress beam is shown in Figure 1. The second kind of beam in use is a typical cantilever beam. This is a beam which has one clamped end and one free end, on which the load is applied.

Figure 1. Constant Stress Beam.

Two different types of strain gage sensors as used in this laboratory. Two single axis strain gages are employed to measure both axial and transverse strain in the constant stress beam. For better accuracy when measuring strain in multiple directions, a strain rosette is used. An example of a rectangular strain gage rosette sensor is shown in Figure 2.

Figure 2. Schematic of a rectangular strain gage rosette.

Both strain gage elements measure strain by the change in electrical resistance, Equation 1, where R is the resistance and F is the gage factor.

[1]

When strain gage output is measured using a Wheatstone bridge, the voltage reading is proportional to the change in resistance (Equation 2), which is directly related to the strain from Equation 1.

[2]

Using this rosette, one can determine the principal strains using geometry and the strain transformation equations. Thus strains recorded by the three gages can give the principle strains of the material, using Equation 1.

[3]

Once the principal strains are known, using the material properties, one can also calculate the stresses using Equation 2 a and b, where E is the Young’s modulus and n is the Poisson’s ratio for the material. Then, using equations from the theory of cantilever beams, we can compare the measured stresses to those calculated using standard formulas.

[4a,b]

RESULTS AND DISCUSSION

Axial and transverse strain measurements were recorded from constant stress beams made of steel, brass, and aluminum. Strain is recorded using a zeroed Wheatstone bridge/digital volt meter as well as a strain gage indicator (SGI). Example data are shown in Figure 3 for the aluminum constant stress beam.

There is good agreement between the two measurement methods for the transverse strain, but there is significant difference between the two methods for the axial strain. This could be due to drift in the Wheatstone bridge or in the strain gage. However, the strain recorded increases linearly with applied load as expected from materials theory. No significant hysteresis is present between loading and unloading the beam.

Figure 3. Axial and transverse strain for a constant stress aluminum beam.

Comparison of the axial and transverse strains from the constant stress beams gives the material’s Poisson ratio. These strains are plotted in Figure 4 and the negative slope of the line is equal to the Poisson ratio. For mild steel, the Poisson ratio is 0.28. For brass and aluminum, the Poisson ratios are 0.27 and 0.33 respectively. These values agree fairly well with the theoretical ratios of 0.27-0.3 for steel, 0.33 for aluminum, and 0.34 for brass [1]

Figure 4. Comparison of axial strain to transverse strain to determine Poisson’s ratio of several materials.

Using cantilever beam theory, we can calculate the stress in the constant stress beam and compare that to the axial strain. The slope of the line from this plot gives the Young’s modulus of elasticity (E), Figure 5. For mild steel, E is found to be 155 GPa. For brass and aluminum, it is found to be 62.5 and 45.6 GPa, respectively. The values are about 30% lower than the accepted values (200 GPa for steel, 100 GPa for brass, and 70 GPa for brass). The source of this error could be the position of the clamp holding the beam to the table during the experiment. The clamp should have been as close to the edge of the table as well as to the point of the beam where the rectangular base meets the triangular shaped section of the beam. This idealized setup what not achieved in the lab and probably led to the discrepancies between the calculated and theoretical values.

Figure 5. Calculated axial stress versus measured axial strain to determine Young’s Modulus for several constant stress beams.

The second part of this study looked at the stress in a simple cantilever beam. A strain rosette (as pictured in Figure 2) is used to measure three strain values at 45 degrees from each other for a beam of aluminum with dimension (9 x 1 x 0.1 inches) that is deflected a known amount (d) using a micrometer. Principal strains and stresses are then computed using Equation 3 and 4a,b and are shown in Table 1. Very good agreement is seen between the calculated principal stress and the calculated stress in the cantilever beam.

Table 1. Measured and calculated stresses and strains of a cantilever beam.
d / eA / sA / eB / sCantilever
mm / mm/mm / MPa / mm/mm / MPa
meas. / calc / meas / calc
0 / 2.41E-06 / 0.2 / -4.14E-07 / 0
2.5 / 1.61E-04 / 11.9 / -4.22E-05 / 11.8
5 / 3.23E-04 / 23.7 / -8.68E-05 / 23.6
7.5 / 4.85E-04 / 35.6 / -1.31E-04 / 35.4
10 / 6.47E-04 / 47.4 / -1.76E-04 / 47.2

CONCLUSIONS

Strain gages were used to measure axial and transverse strains in constant stress and standard cantilever beams. Measured values for Young’s modulus and Poisson’s ratio were close to accepted values; however, errors in the way the beam was clamped may have caused these differences. Strain measurements were made with a Wheatstone bridge/Digital Voltmeter system as well as a strain gage indicator. Generally, the SGI was more accurate. For the cantilever beam, measured principal stains were in excellent agreement with theoretical values.

REFERENCES

[1] Gere, J.M., Mechanics of Materials, 5th ed, Chapman and Hall, London, 2000.

1

Your Name Here TA Name

[1] Corresponding Author