Experiment 5 - Lab Report 5 PDF

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Lab Report 5...

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Lab Report 5 Deflection of Beams

ENGR 244 – MECHANICS OF MATERIALS ONLINE LABORATORY

2. Table of Contents 2. Table of Contents ………………………………………………………………..2 3. Nomenclature …………………………………………………………………..3 4. List of Tables …………………………………………………………………….3 5. List of Graphs ………………………………………………………………….4 6. Objective ………………………………………………………………………...4 7. Introduction 8. Procedure

…………………………………………………………………...4 ……………………………………………………………………...5

9. Results ………………………………………………………………………….5 Part 1 (SIMPLY SUPPORTED BEAM) ……………………………………...5  Part 2 (CANTILEVER BEAM) ……………………………………………….10  10. Discussion …………………………………………………………………….14 11. Conclusion …………………………………………………………………….16 12. Appendixes …………………………………………………………………...18 Appendix A ……………………………………………………………………….18 Appendix B ……………………………………………………………………….19 References

………………………………………………………………………20

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3. Nomenclature Roman Letters M(x) E I P y(x)

Bending moment at a distance x from the end of the beam Modulus of elasticity Moment of inertia of the cross section about the neutral axis Load The curvature of the deflection curve Greek Letters

ρ

Radius of curvature

4. List of Tables 1 The theoretical and experimental deflections for three different beams....7 2 E for each of the beams tested by substituting the experimental deflection values…………………………………………………………………………………..8 3 The theoretical and experimental deflections…………………………...…10 4 E tested by substituting the experimental deflection values………………11

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5. List of Graphs 1 The theoretical and experimental deflection at mid-span as a function of load for each beam metal……………………………………………………………..9 2 The theoretical and experimental deflection at mid-span for each beam metal……………………………………………………………………………....12 3 The theoretical and experimental deflection at full lenght for each beam metal………………………………………………………………………………13

6. Objective In this experiment, the relationship of the load deflection of a simply supported and cantilever beam is evaluated. Also, the modulus of elasticity for different metals is determined [1].

7. Introduction

8. Procedure The procedure for the current experiment consists of two parts which is the procedure for simply supported beam and the procedure for the cantilever beam. For the simply supported beam, the following procedure is being conducted: 1. The cross section of each beam is measured. 2. The specimen is positioned on the supports of the testing machine. 3. Two deformation gauges need to be prepared to measure the vertical deflection at the center and at the quarter point of the beam span. 4. The beams are loaded in increments of 200 N up to 1000 N. Then the deflections at the middle of the span and quarter-span are recorded.

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5. Described procedure is being repeated for the brass, steel and aluminum beams [1]. For the cantilever beam, the following procedure is being conducted: 1. The length, width and height of the beam is measured, using calipers. 2. The deflection display gauge is set to zero. 3. The load hanger is placed at the free end of the beam at the length “L” from the fixed end. 4. The deflection at x= L and x=0.5L is measured for the load increments. 5. Described procedure is repeated for the other cantilever beam metals [1].

9. Results

Part 1 (SIMPLY SUPPORTED BEAM) 1. The curvature of the deflection curve, y(x), is given by: 1 ρ

=

d2 y dx2 dy

(1+(dx )2

3/2

, therefore the equation of the deflection of the beam at any location to )

derive is M(x) EI

d2 y , dx2

where M − bending moment [N * mm], I − moment of inertia [mm4 ], E − Elastic M odulus [P a]. =

An expression for the elastic curve (deflection curve) of the simply-supported beam, can be derived by conducting the following steps: M(x) EI

=

d2 y dx2

knowing that M=(applied load P*length of the beam (where the load was

applied) x)/2, therefore

Px 2EI

=

d 2y dx2

.

Integrating the previous equation, the following result is obtained: Therefore, knowing that

dy dx

L 2 P (L/2)2 (0)EI= 4

= 1 and x =

Applying known conditions,

dy dx

2

EI = P 4x + c1 .

, unknown coefficients c1 can be found. 2

+ c1 , which shows that c1=- P16L .

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Integrating second time, the result EIy= P4 x3 + c1 x + c2 is obtained (boundaries x=0, * 2

y=0). Therefore, (0)EI=0- P16L 0 + c2. This shows that c2=0. Substituting c1 and c2 2

2

2

2

−3L ) Px − PL into EIy= P4 x3 + c1 x + c2 gives, EIy= P4 x3 − P16L x ⇒ y= 12EI ⇒y= P x(4x . x 16EI 48EI * * From the procedure, it is known that the load is applied at the middle of the span 3 and at the quarter-span. Also, I= bh 12 , where b = width of cross section of the beam [mm], h = height of cross section of the beam [mm]. 3

3

3

Therefore, for brass 3

3

I = 19.42mm*1212.75 mm = 3 354.3 mm4 , x = 455mm/2 = 227.5 mm (f or x = L/2). For x=L/4, x=113.7 mm. 3 3 For steel, I = 19.15mm*1212.83 mm = 3370.3 mm4 3

For aluminium, I =

19.27mm* 12.91 mm3 12

= 3 455.2 mm4 .

2. The experimental deflection for the brass can be calculated using the following method (Load-200[N] and x=L/2): y= 200N* 227.5mm(4552 mm2 −3* 4552 mm2 ) 48*105000MP a*3354.3mm4

= 1.11 mm Note: Values are positive since all the results occur below the x (Load) axis. The experimental deflection for the steel can be calculated using the following method (Load-200[N] and x=L/2): y=

200N * 227.5mm(4552 mm2 −3* 4552 mm2 ) 48*105000MP a*3370.3mm4

= 0 .58 mm

The experimental deflection for the aluminium can be calculated using the following method (Load-200[N] and x=L/2): y=

200N *227.5mm(4552 mm2 −3*4552 mm2 ) 48* 105000MP a* 3455.2mm4

= 1 .62 mm

The following table illustrates the calculated data for three different beams when the load is applied at the middle of the span and at the quarter-span of the beams ( the theoretical and experimental deflections).

3. Table 1: The theoretical and experimental deflections for three different beams.

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4. To estimate E for each of the beams, the experimental deflection values are pasted into the elastic curve formula. The formula is y= 2

the formula is E=

P x(4x2 −3L2 ) 48EI

, and for E,

2

P x(4x −3L ) 48yI

.

Therefore, by pasting the experimental values into the formula the following will be calculated: For brass (x=L/2, P=200N): 200N 227.5mm(227.5mm2 −3 4552 mm2 )

* * E= 48* 1.61mm* 3354.3mm4 For steel (x=L/2, P=200N):

E=

=72877 MPa

200N *227.5mm(227.52 mm2 −3*4552 mm2 ) 48*1.61mm*3354.3mm4

=171255 MPa

For aluminium (x=L/2, P=200N): E=

200N *227.5mm(227.52 mm2 −3*4552 mm2 ) 48*1.61mm*3354.3mm4

=631816 MPa

After the results are calculated the following table is developed: Table 2: E for each of the beams tested by substituting the experimental deflection values.

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5.

Graph 1: The theoretical and experimental deflection at mid-span as a function of load for each beam metal.

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Part 2 (CANTILEVER BEAM) 1. The equation of the elastic curve for a cantilever beam (one fixed end) is

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3

−3Lx ) y= P (x 6EI .

3

The moment of inertia is I= bh12 , therefore, 3

3

the moment of inertia for brass beam is: I= 19.36mm*123.49 mm =68.6 mm^4. 3

3

* 3.35 mm =60 mm^4 The moment of inertia for steel beam is: I= 19.00mm12 3

3

The moment of inertia for aluminium beam is: I= 19.20mm*123.52 mm =70 mm^4 x=L, therefore, x =250mm. x=L/2, therefore, x=125mm P (x 3−3Lx 2) 6EI

2. Using the following formula y= different beams will be calculated.

, experimental deflections for three

0.981N *(250mm3 −3*2503 mm3 ) = 0.71 mm 6*105000MP a* 68.6 mm4 3 3 3 N* (250mm −3* 250 mm ) Load=0.981 N), y= 0.981 = 0.43 mm 6*200000MP a* 60 mm4 0.981N* (250mm3 −3* 2503 mm3 ) = 1 .04 (x=L. Load=0.981 N), y= 6 70000MP a* 70 mm4 *

For brass (x=L. Load=0.981 N), y= For steel (x=L. For aluminium

mm

Table 3: The theoretical and experimental deflections.

3. To estimate E for each of the beams, the experimental deflection values are pasted into the elastic curve formula. The formula is y= 3

the formula is E=

P (x3 −3Lx2 ) 6EI

, and for E,

2

P (x −3Lx ) 6yI

.

Therefore, by pasting the experimental values into the formula the following will be calculated: For brass (x=L, Load=0.981 N): 3

3

*2*250 mm =114586 MPa E= 0.981N 6*1.22mm*68.6mm4 For steel (x=L, Load=0.981 N):

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3

*2*250 mm =131010 MPa E= 0.981N 6 0.63mm 60mm4 *

*

For aluminium (x=L, Load=0.981 N): 3

3

*2*250 mm =51043 MPa E= 0.981N 6 1.43mm 70mm4 *

*

After the results are calculated the following table is developed: Table 4: E tested by substituting the experimental deflection values.

4. Graph 2: The theoretical and experimental deflection at mid-span for each beam metal.

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Graph 3: The theoretical and experimental deflection at full lenght for each beam metal.

10. Discussion 1. There

are generally 4 main variables that determine beams deflections, which include [4]: 13

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● ● ● ●

Loading which was applied on the structure The length of the unsupported member Young’s Modulus Moment of Inertia (I) [4].

The accuracy of the theoretical calculations depend on these 4 factors. If one of the points contains a mistake in the solution, the whole result will change (y= ).

The error percentage can be calculated using Error=

yexp −ytheor ytheor

P x(4x2 −3L2 ) 48EI

*100%

Example of beam can be the values from Table 1(x=L/2, Load 200 N), where Error%= 1.61−1.11 *100%=45% 1.11 For steel beam can be the values from Table 1(x=L/2, Load 200 N), where

Error%=

0.68−0.58 0.58

*100%=17%

For aluminium beam can be the values from Table 1(x=L/2, Load 200 N), where Error%= 1.78−1.62 *100%=9.9% 1.62

2. The calculated elastic values, deduced using the elastic curve equation P x(4x 2−3L 2)

P (x3 −3Lx2 )

y= 48EI (for simply supported beams), and y= 6EI (for cantilever beam) slightly differ from the published values, and there are several possibilities of this difference. The most probable is that the loading of the beams was not precise. While the differences are not dramatic, still it is clear that mistakes occur more often while calculating E for simply supported beams, rather than for cantilever beams.

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Another important factor which has to be taken into account when calculating deflection is the nature of the material. It is known that force cannot be applied to a material without changing its dimensions [3]. The greater the value of E, the stiffer is the beam, i.e. the greater is its resistance to being bent [3].

11. Conclusion

12. Appendixes Appendix A

Appendix B

References [1] ENGR 244 Lab Manual 5, 2020. Concordia University. [2] V. Vijay, “Deflection of Beams Study Notes for Mechanical  pdated August 21, 2019. [Online]. Available: Engineering.” gradeup.co. U https://gradeup.co/deflection-of-beams-i-ae206f65-bea5-11e5-b01a-53c804be557a. Accessed on August 1, 2020.

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 pdated February 15, 2015. [Online]. [3] “Beams.” jfccivilengineer.com. U Available: http://www.jfccivilengineer.com/beams.htm . Accessed on August 2, 2020. [4] “What is Deflection?” skyciv.com. [Online]. Available: https://skyciv.com/docs/tutorials/beam-tutorials/what-is-deflection/ . A  ccessed on August 2, 2020.

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