Mechanics of Solids II Experiment D Castigliano\'s Theorem Fall 2021/2022 PDF

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mechanics of solids II experiment D castigliano's theorem fall 2021/2022 lab report...

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MAAE3202B Mechanics of Solids II

Experiment D – Castigliano’s Theorem

25/11/2021

Objective The main objective of this experiment was to measure the horizontal and vertical deflections of four different curved bars subjected to bending and then comparing these results with the theoretical values predicted by Castigliano’s theorem.

Nomenclature

Symbol

Definition Vertical Deflection (Th.) Horizontal Deflection (Th.) Second Moment of Inertia Base Height Radius Horizontal Length Vertical Length Vertical Deflection (Exp.) Horizontal Deflection (Exp.) Applied Load

I b h R s d v u W

Units mm mm mm4 mm mm mm mm mm mm mm N

Experimental Setup and Procedure The experiment was conducted in its entirety in accordance with the outline in the MAAE3202 Lab Manual without any deviations [1]. Dial Gauge (Vertical) Dial Gauge (Horizontal)

Load (Hanger) Curved Bar

Clamp

Figure 1: DIDACTIC Deflection of Curved Bars Apparatus [1]

Figure 2: Bar Geometry for Bars 1-4 (dimensions s, R, and d can be found in Appendix A)

Results and Discussion 1.8 1.6 1.4

Deflection (mm)

1.2

Bar 1 Linear (Bar 1) Linear (Bar 1) Bar 2 Linear (Bar 2) Bar 3 Linear (Bar 3) Bar 4 Linear (Bar 4)

1 0.8 0.6 0.4 0.2 0

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Load (N)

Figure 2: Vertical deflection plotted against the applied load

0.9 0.8 0.7

Deflection (mm)

0.6 0.5

B ar 1

0.4 0.3 0.2 0.1 0

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Load (N)

Figure 3: Horizontal deflection plotted against the applied load

Table 1: Experimental values for the vertical and horizontal deflection per unit load Bar Vertical Deflection per unit load, v Horizontal Deflection per unit load, u (mm/N) (mm/N) 1 0.222 0.145 2 0.183 0.161 3 0.0561 0.102 4 0.306 0.161

Table 2: Theoretical values for the vertical and horizontal deflection per unit load Vertical Deflection per unit load, v Horizontal Deflection per unit load, u Bar (mm/N) (mm/N) 1 0.255 0.120 2 0.185 0.118 3 0.0525 0.0588 4 0.313 0.118

8

Table 3: Percentage Error between the theoretical and experimental values of deflection per unit load

Bar 1 2 3 4

Vertical Percentage Error (%)

Horizontal Percentage Error (%)

13.16727027 0.616792836 -6.86807256 2.430764526

-21.43835 -36.59693 -73.45642 -36.59693

Bar 4 experienced the highest vertical deflection per unit load (0.306mm/N) as well as the highest horizontal deflection per unit load (0.161mm/N). This is due to the shape of bar 4 (Figure 2). This is due to the fact that bar 4 does not have a curved surface whereas bars 1/2 and three have a curved surface on them which allows them to would stand bending due to the fact that the curve is more flexible than the straight corner all the beams experience horizontal a horizontal deflection due to the fact that when the beam is experiencing a vertical load it causes it to bend downwards and eventually that cause it to experience horizontal deflection as well. As for Castigliano’s theorem, a dummy load was used in the derivation of the horizontal and vertical deflection equations in order to calculate the deflection and rotation at a point where there is no force being applied. Castigliano’s theorem also only applies to linear elastic systems such as the one presented in this experiment. Looking at Table 3, it can be seen that the experimental vertical deflection per unit load values were quite close in comparison to the theoretical values, with bar 2 having the smallest percentage of error (0.62%). On the other hand, the experimental horizontal deflection per unit load values were much smaller than the theoretical values. The percentage error ranged from 21% to 73%, meaning that there was a recurring source of error impacting the horizontal results. This could either be a result of not zeroing the gauges between each trial, not making sure that the gauge was perpendicular to the load, parallax error when recording the measurements from the gauges, and not tightening the gauges enough when assembling the load hook. This experiment can be improved by using smaller increments of weights (100g instead of 200g), repeating each trial more than once to obtain an average, and using digital gauges to avoid having a parallax error.

Conclusion It can be concluded that Castigliano’s theorem is able to predict the experimental vertical deflections of a curved bar within a small margin of error. As for the horizontal deflection, it is unclear whether the high percentage of error was from the experiment itself or from Castigliano’s theorem.

References [1] Department of Aerospace and Mechanical Engineering, “Laboratory Experiments Manual”, MAAE3202 Mechanics of Solids II, Carleton University. [Online] [Accessed November 18, 2021].

Appendices Appendix A: Data Tables Bar 1 Mass (g)

Vertical Deflection (0.01 mm)

200 400 600

Bar 2

Vertical Horizontal Horizontal Deflection Deflection Deflection (0.01 (0.01 mm) (0.01 mm) mm)

43 87 130

28 57 85

48 78 120

Bar 3 Mass (g)

Vertical Deflection (0.01 mm)

200 400 600

10 20 32

18 51 81 Bar 4

Vertical Horizontal Horizontal Deflection Deflection Deflection (0.01 (0.01 mm) (0.01 mm) mm) 15 44 21 29 110 55 55 164 84

Specimen

s (mm)

d (mm)

R (mm)

1

75

75

75

2 3 4

0 0 150

0 75 150

150 75 0

Appendix B: Sample Calculations: Second moment of Inertia for the cross-section shown in Figure :

Theoretical vertical deflection, v:

Where W is the applied load, E is the modulus of elasticity, I is the second moment of inertia. “s”, “d”, and “R” and the dimensions of the curved bar as shown below:

Sample calculation for bar 1 subjected to an applied load of 600g:

Theoretical horizontal deflection, u:

Sample calculation for bar 1 subjected to an applied load of 600g:

Percentage Error: Sample calculation for bar 1 vertical deflection per unit load:...