Stress Analysis of Beams Using Strain Gauges

1. Objective

The objective is to determine how different areas along a cross section of a beam act when under pure bending. Then determining the material properties of the beam and comparing them with theoretical values.

2. Introduction

Outside of a laboratory, pure bending rarely happens without another sort of stress such as torsion, shear etc… However, in this experiment, the conditions are set so that we can evaluate a steel beam while it is in pure bending with the objective of observing the different stresses at different locations in the beam. What can be expected is both positives and negative stresses, the reason being, when one side is bent inwards (negative strain, which leads to negative stress) the opposite side of the neutral axes is stretched (positive strains, leading to positive stresses). Another observation that should be made, is as the distance from the neutral axes increases, so will the value of strains observe for the same load, refer to the figure below to better illustrate what the neutral axis is and how to determine it. As a general note, sample calculations will be presented during the presentation of the result to better illustrate how the results were achieved.

Figure 1: Neutral axis diagram

3. Procedure

The procedure for this experiment is quite short. First it is essential to measure the width of the beam, the height and the length between the two loading points. To do so, measure between the inner points of the loading mechanism and the outer mechanism and taking the average of the two. Turn on the strain indicators and reset them to zero with a +/- of 2, after ensuring that there is not a load applied to them. Now turn on the load indicator and reset it to zero, after ensuring that the pump is not applying a load. Now load the beam in increments of 1000N until 5000N is reached. Take photos of all the strain indicators at each interval of 1000N. Once 5000N is reached and the photo has been taken, unload the beam and take a photo of the strain again.

4. Results

4.1 Moment of Inertia

4.2 FBD & Shear/Moment Diagram

4.3 Calculate Bending Stresses At The Strain Points

4.4 Stress-strain curve for gauges 1 and 5

4.5 Longitudinal strain εL for gauge 5 vs load & transverse strain εT vs load

4.6 Experimental stresses using 200 GPa as E

4.7 Shear modulus

4.8 Strain vs distance for load of 5000 N

5. Analysis 

The difference in location will affect if the stress is positive or negative. A negative

stress indicates that location of the gauge is in compression, a positive stress indicates

that the location of the gauge is in tension. The stress is also a function of the distance

from the center line of the beam. The farther away from it, the greater the stress is.

            Discussion question 1, yes the plotted graph of strain vs distance from the mid-plane is approximately linear. This confirms that a plane section before bending remains a plane section after bending. The reason for the graph is not precisely linear is because of variations in results due to machine precision. The strain gauges may not be the most accurate or may have flaws due to age that lead them to have inaccurate results.

            The discrepancies between theoretical and experimental stresses are that the theoretical values do not include any type of possible errors. For example, the bar can have cracks in it that lead to skew experimental results. The temperature in the room when conducting the experiment could have led to the bar being more stiff or less stiff, depending on the temperature. Although theoretical values do sometimes calculate the expected values with regard to a certain room temperature, in the experiment manual it was not stated what that temperature was for the modulus of elasticity being 200 GPa for the steel bar, therefor we can not compare it to the room temperature during the experiment, which was also not measured.

            As observed in table 2 and 3 the strain values are 1 000 000 times bigger, this is because in figure 4 the slopes were needed and when plotted in excel with the values from table 2 the slopes where very inaccurate. Once the strain was multiplied by one million the values where shown as decimals rather than scientific notation, which lead to a more accurate calculation when taking the ratios of gauge 6 and gauge 5. Due to it being a ratio, the multiplication of one million does not affect the results, which was the Poisson’s ratio.

            When observing the experimental bending stresses (Table 1) and the theoretical bending stresses (Table 5) there isn’t a large variation in results. Channel 3s theoretical values not being zero is because the strain obtained is not zero. Although it should be, because channel 3 is on the neutral axis, this is clearly not the case. The cause can be because the strain gauge was not accurately placed on the neutral axis. The reason, channel 6s values vary widely from theoretical to experimental values is because the experimental are transverse stresses whereas the theoretical values was calculated for longitude stresses. Another note is, table 5s results is dependant on the strain from table 2, whereas the values in table 1 are not.

            Figure 3 indicate the stresses of gauge 1 and 5, their absolute values were taken and plotted with respect to strain. They should be parallel and coincident since both the gauges are placed at the same distance from the neutral axis. Gauge 1 is in tension and the other is in compression, this is why their absolute values were taken of both the strain and stress so the graph can simply be in the first quadrant. Their slopes represent the modulus of elasticity of the steel beam in MPa and are very close to the expected value of 200 000 MPa.

            The Poisson ratio obtained is 0.305 for the steel beam whereas the expected value can be between 0.28 and 0.33[1], therefor the obtained results is accurate. The expected shear modulus of steel is 75 GPa [2], the determined modulus was 77.63 GPa.

            The distance in figure 5 when strain is zero is almost zero. This indicates that the neutral axis is shifted by a very small value.

6. Conclusion

In conclusion, the results obtained for the mechanical properties of the steel bar are within a reasonable range of the expected values. This leads to the conclusion, that since those mechanical properties’ calculations were dependent on the values obtained during the experiment , the experiment was therefor conduced successfully.