Unsymmetrical Bending

Summary: The principle axis of an L beam was found experimentally and theoretically and the results were found to accuracy of 7.3%.

Contents page

1. Introduction 22. Apparatus 23. Experiment 34. Theory 34.1Assumptions 34.2 Unsymmetrical Bending Equations 44.2.1Solid edge calculation 65. Results 76. Analysis 76.1Theoretical Error 76.2Empirical error 76.3Accuracy 77. Discussion 88. Conclusions 89. Appendix 89.1 Sample number crunching 99.2 Dimension 99.3 Data 9

1. IntroductionThe resulting deflection on a cantilever L beam for different orientations around the longitudinal axis were found.

2. ApparatusThis experiment consisted of the following items, some are depicted on the next page (Figures 1 and 2).

A built in end with bearing Protractor mounted on the bearing end 1 Kg weight 2 magnetic clock gauges one on the horizontal and one on the vertical plane L beam Clock gauges Micrometer

2

Figure 1 (left) Figure 2 (below)

Weight

L beam

Clock gauges

3. ExperimentThe L beam was set to an initial angle of zero. The micrometer was used to measure the dimensions used for the theoretical calculations. These were taken three times along the length and the average value used. The 1 Kg weight was applied and the resulting deflections measured on the clock gauges were recorded. The beam was then rotated 10 degrees and the corresponding deflections were recorded. This was repeated through 180 degrees. The results were graphed and the principle angle was determined.

4. Theory 4.1 Assumptions

The following assumptions apply to unsymmetrical bending theory of an L beam.Assumption 1 The loads small enough are such that there is no significant axial or torsional deformation.

3

Assumption 2 Squashing action is significantly smaller then bending action. Assumption 3 Plane sections before deformation remain plane after deformation.Assumption 4 Plane perpendicular to the axis remain nearly perpendicular after deformation.Assumption 5 Strains are small.Assumption 6 Material is isotropicAssumption 7 Material is elastic.Assumption 9 There are no inelastic strain. Assumption 10 The material is homogenous across the cross-section.Assumption 11 The average dimensions accurately approximate to those of an equivalent ideal beam.Assumption 1-10 are courtesy of: http://www.me.mtu.edu/~mavable/MEEM4150/Slides/Chapter6.pdfThese assumptions mean that the theoretical beam only deflects smoothly and in the directions predicted by the theory. These assumptions are reviewed in the discussion section. 4.2 Unsymmetrical Bending EquationsThe principle axis is where the neutral axis coincides with the axis of the moment being applied. The principle axis for the x and y component has been calculated as follows. In equations the subscripts refer to the axis and member. e.g. refers to the y axis 2nd member. Otherwise nomenclature is as per notes.The following approach was undertaken to calculate the principle axis.Centriods Second moments of areaPrinciple axis equation

Figure 3

Centroid:

=44.13 6.41=282.9

4

62.70

!11wdf1

6.5310

6.41

44.1312

C

X

Y

X

Y

1

2 .

.

=19.84

Second moments of area:

For a rectangle I around the centre is .

Using parallel axis theorem the I for each rectangle is found and added. The results are tabulated below (Figure 4).

16 177 839 373 855 550 124 191 55 692

179 883

42 480 46 048117 638 186

Figure 4Principle axis equation

(227.2)

=90/2=45

θ=180+45=225

Neutral axis

117 518 204

-116482

=-163

=90 degrees

5

4.2.1Solid edge calculation

Figure 5 An illustration of the L beam in the position of the neutral axis.

This was prompted by the negative value for the which doesn’t make sense. An L beam of the correct geometry was formed. Solid edge calculated the physical properties and then these were used to give the angle of the neutral axis.

Figure 6 Property tableX coordinate 0.53Y coordinate 0.84

32.2 degrees

=0.723

6

=36degrees

5. ResultsFigure 7 shows the experimental results. The table of results is included in section 9.3 Data.

L beam Deflections

-25

-20

-15

-10

-5

0

5

-50 0 50 100 150 200

Rotation (degrees)

x D

efle

ctio

n (

0.01

mm

)

x Deflection

y Deflection

Figure 7 L beam deflections graph

6. Analysis6.1Theoretical ErrorThe theoretical error in calculating the neutral axis was due to the measured geometry error. The dimensions were measured to a precision of ±0.005mm with a micrometer. These errors should be combined according to standard derivative error treatment. The theoretical error was not calculated but qualified in the discussion section

6.2Empirical errorThe empirical error for deflection (±0.005mm) was due to the level of precision of the clock gauges. Rotation was measured with a protractor of ±0.5 degrees precision. The empirical error was included for the graph (Figure 4). This had the limited use of being able to visually show the magnitude of the errors. To make it more useful theoretical defection would also have be plotted. It is possible to deduce that the theoretical value for the principle axis of 36 degrees coincides (within empirical error margins) with the actual value of 30 degrees. 6.3AccuracyThe Accuracy was given by:

This is a reasonable accuracy of 7.3% (the minus sign is irrelevant).

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7. DiscussionThe experiment was reasonably accurate (<10% discrepancy between theory and actual). The both of the graphs are clearly sinusoidal; however in particular, the y deflection graph shows some spurious results. In order to identify the rogue values, best fitting sine curves should be superimposed. Unfortunately the Excel software doesn’t have this function. Alternatively, the moment applied by the weight could be calculated (after measuring the length of the beam) and the theoretical defection computed. This could then be graphed and compared. Some of the errors are likely to be human. This could be reduced by repeated experiments. The assumptions 6, 10 and 11 are slightly dubious. This is because the beam is old. Over the years it has been corroded and possibly permanently deformed somewhat. All the assumptions that rely on the fact that the deflection is insignificant should be valid since the maximum deflection is 3 orders of magnitude less according to a back of the envelope calculation. Unfortunately a mistake that can be traced back to the calculation of the x centriod caused the angle of the principle axis to differ between the two methods. It is unclear why the calculated value is the wrong value of 12 compared to a 14 given by solid edge. This small difference was multiplied throughout the calculations resulting in a 54 degree difference. This illustrates the sensitivity of the experiment to errors. All the theoretical errors have not been quantified; this acts as a useful qualifier.

8. ConclusionThis experiment proved that the simple theory of unsymmetrical bending is valid (accuracy of 7.3%). The results are within experimental error bounds. The experiment could be as mentioned in the discussion.

9. Appendix9.1 Sample number crunching

Average: (62.63 + 62.91 + 62.55) / 3 = 62.6966667

Centroid: (((409.4 * 62.7) / 2) + (282.9 * (6.41 / 2))) / 692.3 = 19.848887

(409.4 + (6.53 / 2) + (282.9 * (6.53 + (44.13 / 2)))) / 692.3 = 12.2810783

: 282.9 * ((44.13 / 2) - 12.28) * (19.84 - (6.41 / 2)) = 46 048.6161

:

((-2) * 117 638 186) / (855 550 + 179 883) = -227.225105

sqrt((((855 550 - 179 883) / 2)^2) + (1.17E8^2)) + 517 716 = 117 518 204

(11 751 804 / (-116 482)) tan(45) = -163.418215

9.1.1Solid edge

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= 0.72degrees

=36degrees9.2 Dimensions    1         2            3Fx   50.81     50.49       50.67Fy   62.63     62.91       62.55t1   6.56      6.22        6.45t2   6.50      6.49        6.59

9.3 Data

Rotation x yx relative error

y relative error

rotation relative error

x absolute error

y absolute error

rotation abs error

0 -4.1 -15.5 -0.00122-

0.00032 0 0.005 0.005 0.5

10 -2 -13 -0.0025-

0.00038 0.05 0.005 0.005 0.5

20 -1 -11.1 -0.005-

0.00045 0.025 0.005 0.005 0.5

30 0 -11.1 0-

0.00045 0.016667 0.005 0.005 0.5

40 0.9 -10.8 0.005556-

0.00046 0.0125 0.005 0.005 0.550 1.7 -12.5 0.002941 -0.0004 0.01 0.005 0.005 0.5

60 3 -13.3 0.001667-

0.00038 0.008333 0.005 0.005 0.5

70 2.8 -14.2 0.001786-

0.00035 0.007143 0.005 0.005 0.5

80 3 -17.1 0.001667-

0.00029 0.00625 0.005 0.005 0.5

90 2.5 -18 0.002-

0.00028 0.005556 0.005 0.005 0.5

100 1.8 -19.1 0.002778-

0.00026 0.005 0.005 0.005 0.5

110 0.8 -18.9 0.00625-

0.00026 0.004545 0.005 0.005 0.5

120 -0.4 -19.9 -0.0125-

0.00025 0.004167 0.005 0.005 0.5

130 -1.9 -18.1 -0.00263-

0.00028 0.003846 0.005 0.005 0.5

140 -2.9 -17.9 -0.00172-

0.00028 0.003571 0.005 0.005 0.5

150 -3.2 -16.1 -0.00156-

0.00031 0.003333 0.005 0.005 0.5

160 -4.3 -15 -0.00116-

0.00033 0.003125 0.005 0.005 0.5

170 -3.8 -13.9 -0.00132-

0.00036 0.002941 0.005 0.005 0.5

180 -3.9 -12.09 -0.00128-

0.00041 0.002778 0.005 0.005 0.5

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