LAB Report 1 ( Bending IN BEAM) PDF

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Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2 014UNIVERSITI TEKNOLOGI MARA FAKULTI KEJURUTERAAN MEKANIKAL Program : Bachelor of Engineering (Hons) Mechanical (EM220/EM221) Course : Applied Mechanics Lab Code : MEC 424 Lecturer : Dr Anizah Kalam Group : Group 1 (EMD4M8B2) MEC 424 - LABORATORY REP...

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Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

UNIVERSITI TEKNOLOGI MARA FAKULTI KEJURUTERAAN MEKANIKAL ___________________________________________________________________________ Program : Bachelor of Engineering (Hons) Mechanical (EM220/EM221) Course : Applied Mechanics Lab Code : MEC 424 Lecturer : Dr Anizah Kalam Group : Group 1 (EMD4M8B2) ___________________________________________________________________________

MEC 424 - LABORATORY REPORT TITLE

:

Bending in Beam

No 1.

NAME AIMI NADIAH BINTI MOHAMAD SAUFI

2.

DANISH DANIAL BIN HASRAN

2020452716

3.

MUHAMMAD AFIF SHAHIMAN BIN SHAHARUDDIN

2020869482

4.

MUHAMMAD HAZMAN HADI BIN HASHIM

2019893102

5.

NUR AINA NAJWA BINTI NORHISHAM

2019892796

LABORATORY SESSION

:

Strength Lab (DATE)

REPORT SUBMISSION

:

23 April 2021 (DATE)

STUDENT ID 2019423182

*By signing above you attest that you have contributed to this submission and confirm that all work you have contributed to this submission is your own work. Any suspicion of copying or plagiarism in this work will result in an investigation of academic misconduct and may result in a “0” on the work, an “F” in the course, or possibly more severe penalties.

Marking Scheme No

1

2

3

4

5

6

7

8

Total

ABSTRACT

Bending in beam, bending characterize the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. When a beam is subjected to a loading system or by a force couple acting on a plane passing through the axis, then the beam deforms. In simple terms, this axial deformation is called as bending of a beam. Due to the shear force and bending moment, the beam undergoes deformation. These normal stresses due to bending are called flexure stresses. In this experiment, we want to determine the elastic modulus (E) of the metals beam using deflection of Mild Steel, Aluminum and Brass method. This method was done by measured the deflection on the middle of the beam using dial gauge when the load is applied continuously. The deflection of the beam due to applied load at two points along the beam will be measured and the elastic modulus is calculated. Then, we want to validate the data between experimental and theoretical values. The results of the experiment might be slightly different from the actual value due to the error that occur during the experiment. We can conclude from this experiment; the experiment can be considered as successful as the values does not have a big difference to the theoretical value.

TABLE OF CONTENTS No. 1

2 3 4 5 6 7

Content Title Abstract Table of Contents List of Tables List of Figures Introduction Theory Experimental Procedures Result Discussion Conclusion References Appendices and Raw Data

Page 1 2 3 4 5 6 7 10 11 26 30 33

INTRODUCTION

Bending is a flexible process which have many different shapes can be produced. In pure bending, we will analysis the stresses and strains in prismatic members subjected to bending. Bending is a major concept used in the design of many machine and structural components, such as beams and girders. An example of pure bending is provided by the bar of a typical barbell as it is held overhead by a weightlifter. The bar carries equal weights at equal distances from the hands of the weightlifter. Because of the symmetry of the free body diagram of the bar in figure 1, the reactions at the hands must be equal and opposite to the weights. Therefore, as far as the middle portion CD of the bar is concerned, the weights and the reactions can be replaced by two equal and opposite 960-lb in. couples in figure 2, showing that the middle portion of the bar is in pure bending.

Figure 1

Figure 2

Bending characterizes the behavior of a slender structural element subjected to external load applied perpendicularly to a longitudinal axis of the element. In this experiment, the beam is subjected to pure bending at the central section. The bending moment is constant and shear force is zero. The maximum deflection y at the midspan of the beam will be measured. The elastic modulus of the beam’s material will be determined from the flexure formula E = M R / I. The objective of this experiment is to determine the elastic modulus (E) of beam specimen by method of deflection of Mild Steel, Aluminum and Brass and to validate the data between experimental and theoretical values.

THEORY

Beams A beam, in Structural Engineering terms, is a member that can be comprised of a number of materials (including steel, wood aluminum) to withstand loads – typically applied laterally to the beam axis. Beams can also be referred to as members, elements, rafters, shafts, or purlins. It is a structural element which is designed and used to bear high load of structure and another external load. There are many different types of beam like cantilever beam, simple supported beam and overhanging beam. Bending of Beam When an external load or the structural load applied in beam is large enough to displace the beam from its present place, then that deflection of beam from its resent axis is called bending of beam. When a beam experiences a bending moment, it will change its shape and internal forces will be developed. Bending Moment Bending moments occur when a force is applied at a given distance away from a point of reference, causing a bending effect. In simple words bending moment is the product of force applied on beam with the distance between the point of application of force and fixed end of the beam. If the object is not well-restrained the bending force will cause the object to rotate about a certain point.

A steel I-beam is subjected to a point load at both ends of it. The beam is loaded within the elastic limit. Simple beam bending is often analyzed with the EulerBernoulli beam equation. The conditions for using simple bending theory are: 1. The beam is subject to pure bending (The shear force is zero, and that no torsional or axial loads are present). 2. The material is isotropic and homogeneous. 3. The material obeys Hooke's law (it is linearly elastic and will not deform plastically). 4. The beam is initially straight with a cross section that is constant throughout the beam length. 5. The beam has an axis of symmetry in the plane of bending. 6. The proportions of the beam are such that it would fail by bending rather than by crushing, wrinkling or sideways buckling. 7. Cross-sections of the beam remain plane during bending.

The elastic modulus of the beam’s material will be determined from the flexure formula; E=

MR I

Where; E: elastic modulus, M: moment, R: radius

I: second moment of inertia. Due to pure bending the beams deflects into the shape of an arc of a circle radius, R; 2

R2=( R− y ) +

() L 2

2

By simplifying and ignoring the small terms; 2

R=

L2 8y

The radius of curvature R of the beam and moment equation is given as; R=

EI EI = M Wd

Moment of inertia on the beam; I=

1 b h3 12

Where, E is the beam’s elastic modulus and I is the moment of inertia about neutral axis. Hence the final formula; E=

( )( d8LI ) W y

2

The pure bending occurred only because of coupling at the ends of the beam which means that there are no shear forces that acts on the cross-section of the beam. In the case for non-uniform bending, the presence of shear forces produces warping or distortion in the cross-section of the beam thus a section that is plane before bending in no longer plane after bending. Usually there are shear forces that acts on the beam, but it is not significantly affects the flexure formula and that we can use it to calculate the normal stresses in cases of non-uniform bending.

EXPERIMENTAL PROCEDURE

1. The Aluminum beam was measured at several places to get the exact dimensions. 2. The width and the thickness of the beam was measured and recorded using a Vernier caliper. 3. After that, the length of the beam was measured and then recorded using a steel ruler. 4. The length, L the position of the weight from the nearest support was set to 100 mm. 5. After the apparatus was setup and the beam were placed on the support. The distance between the 2 support was fixed to 400 mm. 6. The measurement for the dial gauge was fixed and placed at the middle of the beam. 7. The length (X1 and X2) from the wall to the center of the dial calipers was measured and recorded. 8. The dial gauge was set to zero. 9. The weight (W) on the load hanger are hanged at both side of the beam starting from 2N then increasing by increment 2N. 10. The dial gauge then was recorded. The weight (W) was recorded and the deflection y was measured at every increment. 11. The experiment was repeated with several loads 4N, 6N, 8N, 10N, 12N, 14N and 16N. 12. Step 1 to 12 is repeated by replaced the Aluminum beam with brass and mild steel beam.

RESULTS Materials Aluminium Brass Mild Steel Load (N) 0 2 4 6 8 10 12 14 16

Length 998 1005 999

Aluminium 0 0.15 0.32 0.48 0.64 0.80 1.06 1.12 1.28

Width 19.30 20.00 20.36 Materials Brass 0 0.12 0.24 0.35 0.47 0.59 0.70 0.82 0.94

1. AIMI NADIAH BINTI MOHAMAD SAUFI (2019423182)

Thickness 6.52 6.00 4.00

Mild Steel 0 0.19 0.42 0.63 0.84 1.05 1.26 1.47 1.68

Sample Calculations I.

Aluminium Load = 2N Slope

ϕ =

___2N___ 0.15 × 10-3

=

13333.33

First moment of inertia I= = =

bh3 12 (0.0193) (0.00652)3 12 4.458 × 10-10 m4

Young’s modulus of the experiment E= E=

ϕ (X) L2 I × 8 (13333.33) (0.1) (0.4)2

E=

(4.458 × 10-10) (8) 59.838 Gpa

Percentage error Percentage error =

=

| |

|

E Theoretical−E Experimental E Theoritical

(69 )−( 59.838) (69)

|

× 100

× 100

= 13.27 %

II.

Brass Load = 2N Slope

ϕ =

___2N___ × 10-3 0.12

=

16666.66

First moment of inertia bh3 12 = (0.02) (0.006)3 12 = 3.60 × 10-10 m4 Young’s modulus of the experiment I=

E= E= E=

ϕ (X) L2 I × 8 (16666.66) (0.1) (0.4)2 (3.60 × 10-10) (8) 92.6 Gpa

Percentage error Percentage error =

|

|

E Theoretical−E Experiment ETheoritical

× 100

=

|

|

(104.1) −( 92.6 ) (104.1)

× 100

= 11.04 % III.

Mild steel Load = 2N Slope

ϕ =

___2N___ 0.19 × 10-3

=

10526.32

First moment of inertia I= I= I=

bh3 12 (0.0204) (0.004)3 12 1.067 × 10-10 m4

Young’s modulus of the experiment E= E= E=

ϕ (X) L2 I × 8 (10526.32) (0.1) (0.4)2 (1.067 × 10-10) (8) 197.3 Gpa

Percentage error Percentage error =

=

| |

|

E Theoretical−E Experiment ETheoritical

(210 )−( 197.3) (210)

|

× 100

× 100

= 6.04 % Material

Average

Theoretical Elastic

Experimental

Modulus

Elastic Modulus

Percentage of error

Aluminum Brass Mild Steel

55.86 Mpa 94.22 Mpa 177.69 Mpa

69 97 210

19.04% 2.87 % 15.38%

2. DANISH DANIAL BIN HASRAN (2020452716)

Deflection of Beam (mm)

Bending in Beam 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

2

4

6

8

10

12

Load (N)

Chart 1: Bending in Beam Graph Calculation Moment of Inertia I=

1 b h3 12

Where; b = base h = height Elastic Modulus

( )( )

E=

w y

2

dl 8I

Where; E = Elastic Modulus

14

16

18

W= Load y= Deflection l

= Distance between two support

d = Distance between support and load I = Moment of inertia

Percentage of error

|

error=

|

E Experimental−ETheoretical ×100 ETheoritical

Calculation example; Moment of inertia (brass) I brass=

1 3 ( 0.02)( 0.006 ) 12 −10

¿ 3.6 ×10

m

4

Elastic Modulus (brass)

( )(

Ebrass=

16 0.94

(0.01 ) ( 0.04 ) 8 ( 3.6 × 10−10 ) 2

)

¿ 94.56 MPa Percentage of error (brass)

|

error brass= Material

Aluminum Brass Mild Steel

|

55.9− 69 ×100 69

Average

Theoretical Elastic

Experimental

Modulus

Elastic Modulus 55.9 MPa 94.2 MPa 177 MPa

69 97 210

Percentage of error

18.99% 2.89% 15.71%

¿ 2.89 %

Table 2: Calculation table 3. MUHAMMAD AFIF SHAHIMAN BIN SHAHARUDDIN (2020869482)

Bending in beam 1.8 1.6 1.4 Aluminium Brass Mild Steel

1.2 1 0.8 0.6 0.4 0.2 0

0

2

4

6

8

10

12

14

Chart 1: Graph Deflection vs Load

16

Material

Aluminium Brass Mild Steel

Average Experimental Elastic Modulus 56.1 MPa 94.5 MPa 175.4 MPa

Theoretical Elastic Modulus

Percentage of error

69 97 210

18.70% 2.58% 16.48%

Table 2: Calculation table

4. MUHAMMAD HAZMAN HADI BIN HASHIM (2019893102)

Sample Calculation: Aluminium; 1)

Slope from graph:

m=12.145 N /mm ×1000 mm / m ¿ 12145 N /m

2) Finding Moment of Inertia

I=

b h3 12 3

¿

( 0.019)( 0.0065 ) 12

¿ 4.348 ×10

−10

m

4

3) Finding experimental Elastic Modulus, E

E= ¿

m ( x ) L2 I ×8

12145 ( 0.1 ) (0.4 )2

( 4.348 ×10−10) × 8

¿ 55.865GPa 4) Finding Percentage Error

|Theoretical Value−Experimental Value |

Percentage Error= ¿

|69−55.865| 69

Theoretical Value

×100 %

¿ 19.037 %

Brass: 1) Slope from graph

m=17.093 N /mm ×1000 mm / m ¿ 17093 N /m

2) Finding Moment of Inertia

I= ¿

b h3 12

( 0.020 )( 0.006 )3 12

¿ 3.6 ×10

−10

m

4

3) Finding experimental Elastic Modulus, E

×100 %

E=

¿

m ( x) L I ×8

2

2 17093 ( 0.1 ) ( 0.4) (3.6 ×10−10 )× 8

¿ 94.961 GPa 4) Finding Percentage Error

|Theoretical Value−Experimental Value |

Percentage Error= ¿

Theoretical Value

|97− 94.691| 97

×100 %

¿ 2.102 % Mild Steel: 1) Slope from graph

m=9.4776 N /mm × 1000mm / m ¿ 9477.6 N /m

2) Finding Moment of Inertia

I=

b h3 12 3

¿

( 0.02036)( 0.004 ) 12

¿ 1.0859 × 10−10 m 4

×100 %

3) Finding experimental Elastic Modulus, E

E= ¿

m ( x ) L2 I ×8

9477.6 ( 0.1) ( 0.4 ) 2

( 1.0859 ×10−10)× 8

¿ 174.558GPa 4) Finding Percentage Error

|Theoretical Value−Experimental Value |

Percentage Error= ¿

|210−174.558| 210

Theoretical Value

×100 %

×100 %

¿ 16.877 %

5. NUR AINA NAJWA BINTI NORHISHAM (2019892796)

DEFLECTION OF BEAM VS LOAD GRAPH 1.8

DEFLECTION OF BEAM, y

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

2

4

6

8

10

12

14

LOAD, N Materials Aluminium

For aluminium b = 0.0193m h= 6.52 x 10^-3 m I= 4.458 x 10^-10 m^4

Materials Brass

Materials Mild Steel

16

18

For mild steel b =20.36 x 10^-3 m h = 4 x 10^-3 m I = 1.086 x 10^-10 m^4

For brass b = 0.02m h= 6.0 x 10^-3 m I = 3.6 x 10^-10 m^4

1) Moment of Inertia

, I=

2) Elastic Modulus,

E=

3) Percentage of Error,

1 b h3 12

( )( ) W y

dL 8I

|

Error=

2

|

EExperimental − ETheoretical × 100 % ETheoretical

Sample of calculation



Moment of Inertia

For Aluminium, IAL:

I Al =

3 1 ( 0.0193 ) ( 6.52 ×10−3) 12

I Al =4.46 ×10

−10

m

4

For Mild Steel, ISt:

I St=1.086 ×10

−10

m

4

For Brass, IBr : −10

I Br=3.6 × 10

4

m



Elastic Modulus

For Aluminium, IAL:

0.4 ¿ ¿ ¿2 (0.1)(¿¿ 8(4.46 ×10−10)¿) ¿ 16 E= ¿ 0.00128

(

)

E Al=56.05 GPa

For Mild Steel, ISt:

ESt =175.39 GPa For Brass, IBr :

EBr =94.56 GPa Moment of Inertia

For Aluminium, IAL:

|56.0569−69|× 100 %

Error Al =

Error Al =18.77 %

For Mild Steel, ISt:

Error St =16.48 % For Brass, IBr :

Error Br=2.52 %

Material

Average Experimental Elastic Modulus (GPa)

Theoretical Elastic Modulus (GPa)

Percentage of error

Aluminium

56.05

69

18.77%

Brass

175.39

97

2.52%

Mild Steel

94.56

210

16.48%

DISCUSSION

1. AIMI NADIAH BINTI MOHAMAD SAUFI (2019423182)

In this experiment, we run the experiment by using three different materials of metals. We used aluminium, brass and mild steel. The length, width and thickness of these three different metals were measured and recorded using Vernier caliper and steel ruler. When the beams were subjected to bending, values of modulus elasticity or Young’s modulus (E) of the three metals (aluminium, brass and mild steel) were calculated and the results were compared with the theoretical values. Basically, the theoretical value for elastic modulus of aluminium, brass and mild steel are 69 Gpa, 104.1 Gpa and 210 Gpa respectively. From the result obtained, the deflection of all the materials are directly proportional with the load applied. It can be determine that the load increase, the deflection of the material also increases. The experimental value of modulus of elasticity for these materials obtained by the calculatio...