s1: The bending of beams

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SummaryThe structures laboratory consisted of three tests which were used to determine the relationship between the bending moment and the radius of curvature. The first experiment was used to observe the uniform deflection in the beam. The second and third experiments were carried out to see the effects of the bending moment when the formation of loading on the beam is varied. A simply-supported beam was used and the deflection was measured. Graphs were plotted and the gradients were used to calculate the youngs modulus of the mild steel used for the beam. The relationship between the bending moment and radius of curvature was found to be inversely proportional; an increase in the bending moment, meant a decrease in the radius.

Table of Contents

Introduction3

Theory4

Apparatus5

Method6

Calculations and Analysis8

Relationship between M and R13

Discussion14

Conclusion15

References16

Appendix17

Introduction

Beams are structural members which have many uses in engineering, such as buildings and bridges. A simply supported beam is one which is supported from both ends and is free to rotate. Beams typically are utilised to support vertical loads, which cause bending moment. Therefore there is deflection in loaded beams due to different forms of bending moment. As a beam is loaded, it deflects (as shown in figure 1), if the path of deflection was to carry on, a circular arc path would be formed.

The experiments which were performed in the structures laboratory were used to determine the effects which different types of bending moment has on deflection in the beam. There were four main aims:1. To show a loaded beam which has a uniform cross section, tends to bend in a circular arc due to the bending moment.2. To prove that radius of the curvature along the length of the beam, R, changes when the bending moment, M, changes.3. To identify the specific link between R and M.4. To calculate the Youngs Modulus of elasticity, E, for the material which the beam is constructed with.

Theory

The bending equation shows that at any point on a beam

And therefore,

Where R is the radius of curvature, E is the Youngs Modulus, I is the second moment of area relative to a transverse axis thought a neutral axis and M is the bending moment.This formula works for any transverse section of the beam.For a bridge gauge, if the bending over bridge gauge length, l, is circular, then:

And therefore,

However in beams, the value of is small and therefore can be neglected. This means the equation above can be simplified to and further more rearranged to find the radius of curvature,

The illustration of this equation can be seen in figure 2.

Apparatus

Beam (simply-supported) Knife-edged supports Weight plates (0.5 Kg and 5 Kg) Weight hangers Bridge Strain Gauge (Mitutoyo 1013F) Each division = 0.002 mm Separate dial test gauges to measure deflection (Digital and analogue)A beam with a set length of 1300mm was used and was supported from both ends by knife edged supports. Load hangers were attached to the beam at specific positions and the required weight plates were placed upon them. A dial test gauge was used to measure the deflection present in the beam. It was slid across the beam by small intervals towards the centre to see how deflection varies as it gets closer to the centre. The number of increments measured on the bridge gauge was multiplied by 0.002 to get the measurement in terms of mm. The bridge gauge is shown in figure 5.

Method

For all the experiments a mild steel beam was used, which had a cross section of 31.75mm width (B) and 6.35mm depth (D). The loads were chosen so the bending moment did not surpass a value of 11.0Nm. The setup for experiments 1 and 2 is shown in figure 3. However for for experiment 2 the bridge gauge was removed and instead a dial gauge was placed at the centre of the beam.Experiment 11. The bridge gauge was fixed onto the beam.

2. The weights were removed, leaving just the weight hangers, and then the bridge gauge was calibrated to read zero.

3. The weights (3.5kg on each side) were then added and the reading on the gauge was recorded.

4. The weights were removed and the position of bridge gauge was varied by 59mm increments towards the centre. Only half the beam was tested to remove any duplicate readings.

5. The results were tabulated with columns of x, the length from the support to the centre of bridge gauge and 1, the bridge gauge reading caused by the addition of weights.Values obtained shown in Calculations and Analysis section of report (Table 1).

Experiment 21. The weights were removed, as well as the bridge gauge.

2. A separate dial test gauge (digital) was set up in the middle of the entire length of the beam, which could measure the central deflection in the beam.

3. Weights were added at increments of 0.5kg, up to a maximum of 3.5kg and therefore seven readings were made (excluding the initial reading where no weights were attached). The readings for central deflection were recorded.

4. The weights were then unloaded, again in increments of 0.5kg until there were no more weights attached. This was to get a second reading of central deflection of each weight.

5. The results were tabulated with columns for W, Load and 2, the central deflection.Values obtained shown in Calculations and Analysis section of report (Table 2).Experiment 3For experiment 3, a different set up was used as shown in figure 4.1. The bridge gauge was placed on the beam as well as an analogue dial gauge.

2. The loading were removed and the bridge gauge was calibrated to show zero.

3. A weight of 5kg was placed onto the weight hanger and the reading on the dial gauge was recorded.

4. The weight was removed and the readings on the dial gauge was checked to show zero. If otherwise, the measurement would have needed to be repeated due to a calibration error.

5. The bridge gauge was then shifted towards the centre, and for each position 3, the gauge reading for deflection was recorded. The reading was then multiplied by a thousand to make it easier to plot on a graph.Values obtained shown in Calculations and Analysis section of report (Table 3).

Calculations and Analysis

Experiment 1As , if E, I, and M, are all constant, then the equation shows that the beam will bend with a path of a circular arc. X (mm)1No. increments (mm) (x 0.002)

45270.054

104270.054

163270.054

222270.054

281270.054

340270.054

Table 1Table 1 shows as the length from the support to the centre of bridge gauge is varied, the number of increments does not change. Therefore it can be stated that the beam was bending upwards, and the same amount of deflection occurred in all sections of the beam.Experiment 1 was held out to simply observe the fact that the deflection will occur at a constant magnitude, along all sections of the beam.

Experiment 2Moment = Wa at all points on the beam between the supports, where a = 300mm.

Where L is the span of the beam between the supports and the value of L was 700mm. As a value of 2 was obtained, it was possible to calculate the radius of curvature for the beam when each weight was added.If M = Wa and are subbed into this equation, it is transformed into:

Where is the gradient of the graph 2.

As B = 31.75 and D = 6.35,

Load (W)2 Loading(mm)2 unloading(mm)2 Average(mm)Moment (N.mm)(M=Wa)R(mm)(R=L/82)

(kg)(N)( kg x 10 )

0.0000---------------

0.550.690.690.69150088768.12

1.0101.361.381.37300044708.03

1.5152.032.082.06450029733.01

2.0202.722.762.74600022354.01

2.5253.383.453.42750017909.36

3.0304.074.134.10900014939.02

3.5354.734.734.731050012949.26

Table 2Table 2 shows that as the load is increased, the deflection in the centre of the beam, 2 is also increased. A theoretical value of E can be calculated by using the following method: Where the values of M and R can be taken from any row in table 2. The value of I, the second moment of area was calculated before at 677.46 and so,

This is the expected value of Youngs Modulus obtained before graph 2 was plotted for experiment 2. Looking at graph 2, it can be seen that as the load increases, so does the average deflection in the beam. The best line of fit proves the relationship is directly proportional, and a gradient of W/2 can be easily obtained. Gradient = This gradient can be used to check if the obtained Youngs Modulus from the experiment is accurate.

This shows the experimental value is 1.76% bigger than the expected value.Experiment 2 was done to show how deflection is affected when the loads are placed on the ends of a simply-supported beam.

Experiment 3The beam is symmetrical in this experiment and therefore the weight is distributed on each support by W/2. The mass used was 5 kg which means it had a weight of 50 N.Therefore:

Where x is the distance between the support and the centre of the bridge gauge.The radius of curvature is:

Where l is the span of the bridge gauge.These two equations can be substituted into which care therefore be transformed into:

Where is the gradient of Graph 3.X(mm)Moment (N.mm)R(mm)

No. increments(mm)(x 0.002)x 1000

4520.00441125175781.25

10460.01212260058593.75

163100.02020407536156.25

222130.02626555627043.27

281170.03434702520680.15

Table 3The value of I is constant and the values of M and R could be attained from Table 3, and so the theoretical Youngs modulus value could be found:

After Graph 3 was plotted, the gradient could be used to get the experimental value of Youngs Modulus:

Hence,

This means the experimental value of Youngs Modulus of the mild steel was 3.98% smaller than the expected value.Experiment 3 gave results which helped determine the effects on beam deflection by adding a load to the centre of a simply-supported beam.

Relationship between M and R

The experiments carried out show that there is a close link between bending moment and radius of curvature. Specifically, the relationship is of a negative correlation. If the bending moment is increased, then the radius of curvature decreases. This can be seen in the tables of results in experiment 1 and experiment 2 (See Table 1 and Table 2).However, the connection can be understood with more clarity with the aid of Graph 4. The graph has both the data from experiment 2, and experiment 3. The two lines share a very similar path, and at points are almost relatively, identical.When the bending moment is equal to 7000 Nmm, the radius of curvature for the beam in experiment 3 is approximately equal to 21,000 mm. For the beam used in experiment 2, at the same value of bending moment, the radius is roughly 19,000 mm.The gradient of the curves rapidly decrease initially, which suggests the large amount of bending moment was deflecting the beam more, which resulted in a smaller radius of curvature. As the bending moment reaches the range between 2600-3000 Nmm, the gradient of the graph appears decrease at a constant rate. For these reasons it is possible to say there is a direct impact on the radius of curvature, R, when the bending moment, M, is varied.

DiscussionLoad and deflection are directly proportional as shown on the graph 2 by a straight line of best fit. The Youngs Modulus of mild steel, has an expected range of 200,000-210,000 N/mm2. It can also be stated that the experimental value is smaller than the expected value for Youngs Modulus of the mild steel used for the beam in experiment 2, however bigger than the expected value in experiment 3. This can be due to the errors which may have occurred during the practicals as well as the effect of limiting factors, such as inaccuracy of readings on the digital dial gauge for more than two decimal places.

The percentage error for the second experiment is higher than the error for the third experiment, and it proves that experiment 2 was less accurate, which could be due to the use of an analogue dial gauge for the third experiment. Another reason why experiment 2 may have a bigger error, is due to the weight of the bridge gauge. This is because it would mean there is an additional load which is not being accounted for when measuring the deflection in the simply-supported beam.The errors could be caused by defects in the apparatus or precision of the measuring instruments used.A cantilever beam (figure 6), which is subject to a concentrated load on the end which is not supported, will deflect. A bridge gauge can also be used to measure the deflection.

Conclusion

The experiments were successful and allowed a number of different values to be calculated, for example the Youngs Modulus and the radius of curvature. The resulting conclusion is that the bending moment and radius of curvature are inversely proportional. Also it has been found that as load placed on the beam increases, the deflection in the beam also increases. The errors could have been reduced by increasing the precision of the bridge gauge, as well as taking into account the weight of the gauge. More readings could have been taken to have more accuracy.All the aims of the experiments were achieved.

References

Safdar, Z. (2015).Figure 1 [image]

Safdar, Z. (2015).Figure 2 [image]

Safdar, Z. (2015).Figure 5 [image]

Figure 6 - Clyne, B. (2015).DoITPoMS - TLP Library Thermal expansion and the bi-material strip. [Online] Doitpoms.ac.uk. Available at: http://www.doitpoms.ac.uk/tlplib/thermal-expansion/printall.php [Accessed 4 Feb. 2015].

Safdar, Z. (2015).Graph 1 Experiment 1 1 (mm) against x (mm). [Graph]

Safdar, Z. (2015).Graph 2 Experiment 2 Load (N) against 2 (mm). [Graph]

Safdar, Z. (2015).Graph 3 Experiment 3 X (mm) against 3 (x10 -3 mm). [Graph]

Safdar, Z. (2015).Graph 4 Relationship between M and R. [Graph]

Appendix

Figure 1 - (Safdar.Z 2015)

Figure 2 (Safdar.Z 2015)

Figure 3

Figure 4 Figure 6 (Clyne, 2005)

Figure 5 (Safdar.Z 2015)

Graph 1 Experiment 1 1 (mm) against x (mm) (Safdar.Z, 2015)

Graph 2 Experiment 2 Load (N) against 2 (mm) - (Safdar.Z, 2015)

Graph 3 Experiment 3 X (mm) against 3 (x10 -3 mm) - (Safdar.Z, 2015)

M (N.mm)