Beams Flexure Experiment Laboratory Write-Up Report
Short Description
McGill CIVE 207 lab 2 Beams Flexure Experiment Laboratory Write-Up Report...
Description
McGill University Department of Civil Engineering and Applied Mechanics
Solid Mechanics CIVE-207
Beams Flexure Experiment Laboratory Write-Up Report 33 Pages
Group 34 Chemen, Uyirendiren
260732722
Earle, Stuart
260735993
Nottegar, Alex
260707220
Zhang, Qitong
260732944
Sabourin, Christophe
260744128
March 16th, 2018
Table of Contents List of Graphs, Drawings and Figures, Photographs and Videos, Tables …................. p.2 Graphs …........................................................................................................................ p.4 Photographs …............................................................................................................... p.13 p.13 Drawings and Figures ………………………..………..................................................p.21 ………………………..………..................................................p.21 Tables …........................................................................................................................ ... p.26 Discussion ………………………… ……………………………………………………… ……………………………………………...….. ………………...….. p.26 Appendix …................................................................................................................... p.28 p.28 References ……………………………… ………………………………………………………… ………………………………………...…. ……………...…. p. 33
1
List of Graphs, Drawings and Figures, Photographs Photographs and Videos, Tables Graphs
Influence of Young’s Modulus of Elasticity 1. 2. 3. 4.
Applied load (N) vs. Mid-span Mid-span deflection, δmid-span, (mm) Applied load (N) vs. Angle of end rotation, θend, (rad) Applied load (N) vs. 48δ0.5LI/L3 (mm2) Flexural stiffness, k, (N∙mm) vs. Moment of inertia, I, (mm4)
Influence of Support Conditions 5. 6. 7. 8. 9.
Applied load (N) vs. Maximum deflection, δmax, (mm) Applied load (N) vs. Free-end Free-end deflection, δfree-end, (mm) Applied load (N) vs. Angle of rotation, θ, (rad) 1/δmax (mm-1) vs. EI/PL3 (mm-1) Flexural stiffness, k, (N∙mm) vs. Maximum deflection, δmax, (mm)
Photographs
Influence of Young’s Modulus of Elasticity Mild Steel 1. Mild steel test before adding load 2. Mild steel test at maximum load Aluminum 3. Aluminum test before adding load 4. Aluminum test at maximum load Brass 5. Brass test before adding load 6. Brass test at maximum load Wood 7. Wood test before adding load 8. Wood test at maximum load Acrylic 9. Acrylic test before adding load 10. Acrylic test at maximum load 2
Influence of Support Conditions Propped Cantilever (fixed-roller) 11. Propped cantilever test before adding load 12. Propped cantilever test at maximum load Cantilever (fixed-free) 13. Cantilever test before adding load 14. Cantilever test at maximum load Fixed Ended (fixed-fixed) 15. Fixed ended test before adding load Drawings and Figures
Influence of Support Conditions Propped Cantilever (fixed-roller) 1. 2. 3. 4.
Line diagram of beam configuration Elastic curve profile Shear force diagram Bending moment diagram
Cantilever (fixed-free) 5. 6. 7. 8.
Line diagram of beam configuration Elastic curve profile Shear force diagram Bending moment diagram
Fixed Ended (fixed-fixed) 9. Line diagram of beam configuration 10. Elastic curve profile 11. Shear force diagram 12. Bending moment diagram Tables
1. Influence of Young’s modulus of elasticity theoretical vs. experimental values 2. Influence of support conditions theoretical vs. experimental values
3
Graphs
1 h p a r G
4
2 h p a r G
5
3 h p a r G
6
4 h p a r G
7
5 h p a r G
8
6 h p a r G
9
7 h p a r G
10
8 h p a r G
11
9 h p a r G
12
Photographs Influence of Young’s Modulus of Elasticity Mild Steel
Photograph 1: Mild steel test before adding load
Photograph 2: Mild steel test at maximum load
13
Aluminum
Photograph 3: Aluminum test before adding load
Photograph 4: Aluminum test at maximum load
14
Brass
Photograph 5: Brass test before adding load
Photograph 6: Brass test at maximum load
15
Wood
Photograph 7: Wood test before adding load
Photograph 8: Wood test at maximum load
16
Acrylic
Photograph 9: Acrylic test before adding load
Photograph 10: Acrylic test at maximum load
17
Influence of Support Conditions Propped Cantilever (fixed-roller)
Photograph 11: Propped cantilever test before adding load
Photograph 12: Propped cantilever test at maximum load
18
Cantilever (fixed-fixed)
Photograph 13: Cantilever test before adding load
Photograph 14: Cantilever test at maximum load
19
Fixed Ended (fixed-fixed)
Photograph 15: Fixed ended test before maximum load
20
Drawings and Figures Propped Cantilever (fixed-roller)
1Figure
1: Propped cantilever line diagram of beam configuration
2Figure
2: Propped cantilever elastic curve profile
1
Goodno, Gere; 2013. Gere, Timoshenko; 1997.
2
21
2Figure
2Figure
3: Propped cantilever shear force diagram
4: Propped cantilever bending moment diagram
Cantilever (fixed-free)
Figure 5: Cantilever line diagram of beam configuration
22
3
Figure 6: Cantilever elastic curve profile
4Figure
7: Cantilever shear force diagram
3
Beer, Dewolf, Johnston; 2002. http://output.to/sideway/default.asp?qno=120800023
4
23
4Figure
8: Cantilever bending moment diagram
Fixed Ended (fixed-fixed)
5Figure
5
9: Fixed ended line diagram of beam configuration
http://www.structx.com/Beam_Formulas_016.html
24
Figure 10: Propped cantilever elastic curve profile
5Figure
5Figure
11: Fixed ended shear force diagram
12: Fixed ended bending moment diagram
25
Tables Table 1: Influence of Young’s Modulus of Elasticity Theoretical vs. Experimental Values Mild Steel T E E (GPa) Pmax (N)
Wood
T
E
T
E
Acrylic T E
178
70
63.5
100
103
12
5.18
2.8
2.90
20
20
20
20
20
20
20
20
20
20
2.30
2.58
6.41
7.09
5.20
5.02
2.86
6.60
3.53
3.00
0.0172
0.0184
0.0480
0.0710
0.0390
0.0324
0.0570
0.0908
0.106
0.0604
2000
2000
2000
2000
2000
2000
750
750
500
500
-
57.15
-
56.20
-
61.94
-
27.09
-
16.32
(mm) (rad) Mmax (N∙mm) σmax (MPa)
Brass
200
δmax θmax
Aluminum T E
Table 2: Influence of Support Conditions Theoretical vs. Experimental Values
Pmax (N) δmax (mm) θmax (rad) α
Mmax (N∙mm)
Propped Cantilever (fixed-roller) T E 20 20 1.07 1.24 0.0138 0.0102 95.45 1285.2
Cantilever (fixed-free) T
E
20 4.60 0.0345 4000
20 4.76 0.0487 29.11
Fixed Ended (fixed-fixed) T 20 0.58 1000
E 20 0.69 182.81
* Theoretical values obtained from Material Properties Table.
Discussion Comparison of Experimental vs. Theoretical Results One can notice that the experimental results for the wood are double what they should be which can be explained by the fact that the Young's Modulus is halved. This is most likely due to the wide range of Young's Moduli for different types of wood. Also, there is a l ot of variance in wood from sample to sample due to its organic nature. For all the different support conditions for mild steel, the theoretical value is lower than the experimental value. This may be explained by the fact that the beam had been used by previous teams and had become slightly less resistant to flexure than initially. 26
The experimental angle for acrylic is lower than expected, this can be explained by the fact that the dial gauges were pushing downward against the beams, lowering the angle. Having the lowest Young's Modulus, the acrylic's angle should be that which is most affected. The angle would also have been increased by the dial gauge at the center, for all beams but the acrylic, the middle gauge should overcome the effect of the edge gauge due to its bigger distance from the pivot, explaining higher angles for these materials.
Sources of Error This experiment was carried out by following a precise procedure in order to minimize potential error. Many sources of error were accounted for by takin g the initial deflections of the beams, and the results were zeroed to these values. Despite this, some sources of error will be acknowledged and explained here. Those discussed in this section are the defects in beam materials and frequent use of certain materials.
Defects in the materials can range from manufacturer mishaps to damages sustained in the laboratory, and can be very difficult to diagnose. For that reason, material defects has been included as a primary source of error. In reference to results received in this experiment, material defects is the outstanding reason for the large difference between the experimental and theoretical value for the Young’s Modulus of the wooden specimen. As the procedure was followed in a nearly identical manner for all beams, the main reason for this large difference can be deduced as a defect in the wood. This is the rational because wood as a material is cut and sanded to size from trees, which contain knots and ruts, while the other materials are manufactured to size.
Frequent use of the particular beams used in this experiment is also a source of error. This experiment was performed after a series of groups had already completed the same experiment, thus each beam had already experienced the loading several times. After several uses, these materials will not perform as adequately as when they were produced. This means the beams will most likely deflect more than they would have in the first trial, which can be linked to the values of the experimental Young’s Moduli being slightly less than the theoretical values for the materials.
27
Appendix Derivation of Fixed-Fixed elastic curve equation To calculate elastic curve v(x) and θ(x), we start by dividing the displacement into one force induced displacement and two moment induced displacements. Suppose ve ctors going up and right is positive. The displacement v induced by a downward force P along x axis is:
4 3 = 48 ( 0 < < 2) For x>L/2, substitute x = (L-x) into the equation above we get:
84 = (2 < < ) 48 Then M is the same on both side because P acts at the middle point, so we can have v induced by the left side moment is:
3 2 = 6 The displacement, v, induced b y the right side moment is obtained by substitute x with (L-x):
= 6 Then we add all equations up and get:
4 2 22 3 ) = 48 ( 0 < < 6 2 84 2 22 = (2
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