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MECHANICS OF MATERIALS LAB MEC424 BENDING IN BEAM

MR. SUHAIME SUPARMAN

GROUP: EMD4M12A MOHAMAD FIKRI BIN MOHD SHAHIDAN – 2010315603 MOHD SAFWAN BIN ABDULLAH – 2010176119 MOHAMMAD ILIYAS HAKIM BIN IBRAHIM – 2010967927

1

ABSTRACT When a beam is supported at two different point and loads are applied anywhere on the beam, we can mathematically determine the deflection that occurs on the beam. With the deflection know, we can also determine the Young’s Modulus (modulus of elasticity) of the material used in the experiment. The following procedures will explains how the experimental values of the deflection of beam and the modulus of elasticity of the beam are determined.

2

TABLE OF CONTENT 1.0 INTRODUCTION……………………………………………………………………………………………..4 1.1 GENERAL BACKGROUND…………………………………………………………………….4 1.2 THEORY………………………………………………………………………………………………4 2.0 EQUIPMENTS AND PROCEDURES……………………………………………………………………6 2.1 EQUIPMENTS………………………………………………………………………………………6 2.2 EXPERIMENTAL PROCEDURES…………………………………………………………….6 3.0 RESULTS………………………………………………………………………………………………………….7 3.1 TABLE OF RECORDED RESULT……………………………………………………………..7 3.2 GRAPH OF RESULT……………………………………………………………………………….8 3.3 CALCULATION OF RESULT……………………………………………………………………9 4.0 DISCUSSION…………………………………………………………………………………………………..10 5.0 CONCLUSION…………………………………………………………………………………………………11 6.0 REFERENCE…………………………………………………………………………………………………….12 7.0 RAW DATA………………………………………………………………………………………………………12

3

1.0 INTRODUCTION 1.1 GENERAL BACKGROUND If a beam is supported at two different point and loads is applied anywhere on the beam, a deflection of beam will occur. When the load are applied either on the outside of the inside of the support (in this case, the loads are applied on the outside of the support), the deflection of the beam can be mathematically determined by using the beam’s properties and geometries.

1.2 THEORY The figure below is the how the setup of the experiment will look like.

Where; R = Radius of curvature of beam, L = length of the beam, y = the deflection of the beam, x = the distance from the end of beam to support, W = weight of the load,

4

Pure Bending Theory;

(

)

( ⁄ ) ⁄

Therefore:

⁄ ⁄ and;

( ) ⁄

Therefore;

( )

(

) ( ) (

) ( )

Graph of W against y;

5

2.0 EQUIPMENTS AND PROCEDURES 2.1 EQUIPMENTS 1. 2. 3. 4. 5. 6. 7. 8. 9.

Weight Load holders Hanger Universal magnetic stand – to hold the gauge Gauge – to read the deflection Ruler – To measure the length of beam Screw driver Pliers Beam – (mild steel, aluminium, brass) – the beam should be rectangular, long and thin. The exact dimension of the beam should be measured and recorded 10. Vernier Caliper – To measure beam width

2.2 EXPERIMENTAL PROCEDURES 1. The dimension (width, height and length) of all the beams are measured with the vernier caliper and ruler and are recorded. 2. The 2 load holders are placed on the beam (mild steel) at a distances 15cm from each of support bars. 3. The beam is placed on the support bars. 4. The gauge is attached to the universal magnetic stand and is position at the middle of the beam. 5. The reading of the gauge is reset to zero. 6. Loads of 2N are added at each of the load holders simultaneously and an increment of 2N is added until a load of 16N is achieved at each load holders. 7. As the loads are added, the deflection reading from the gauge for each of the load is recorded. 8. Steps 2 to 7 are repeated for the remaining 2 beams (aluminium and brass). 9. The recorded deflections of the beams are then tabulated.

6

3.0 RESULTS Load (N)

Beam, Max Deflection (mm) Mild Steel

Aluminium

Brass

0

0

0

0

2

0.15

0.24

0.15

4

0.4

0.49

0.32

6

0.635

0.735

0.49

8

0.88

0.98

0.66

10

1.125

1.225

0.82

12

1.37

1.47

0.99

14

1.61

1.71

1.155

16

1.85

1.91

1.32

7

8

Calculation: Mild Steel: According to the graph drawn, the slope of the deflection for mild steel is 8431;

[

(

) ] ( )

( [

) (

) (

]

)

Aluminium: According to the graph drawn, the slope of the deflection for aluminium is 8272;

[

(

) ] ( )

( [

) (

) (

)

]

Brass: According to the graph drawn, the slope of the deflection for brass is 12040;

( [

) ] ( )

( [

) (

) (

)

]

9

4.0 DISCUSSION 1. Compare the value of E (modulus of elasticity) obtained from this method with their theoretical value. After conducting this experiment and calculate the data that was recorded, the value of E (modulus of elasticity) of Mild Steel is 220.19GPa, Aluminium is 68.93GPa and Brass is 100.33GPa. Whereas theoretical value of these beams (from text book) such as Mild Steel is 210GPa, Aluminium is 69GPa and Brass is 102GPa.

2. Comment on your result. The calculated E (modulus of elasticity) of the beams is not too different from their theoretical value. Such examples are the experimental value of E of Aluminium is 68.93GPa and its theoretical value is 69GPa. The different of this value is 0.07GPa or 0.10% different. Whereas the experimental value of E of Brass is 100.33GPa and its theoretical values is 102GPa. The different of value is 1.67GPa or 1.64%. Finally the experimental value of E for mild steel is 220.19GPa while its theoretical value is 210GPa. The different between these two values is 10.19GPa or 4.85%. The different in values between experimental and theoretical can be cause by error while conducting the experiment itself. In every experiment there will be error but it is the main goal to minimize these errors. Errors in values that are less that 5% are acceptable. 3. What are the other methods available to determine E of the materials. Other than determining the E (modulus of elasticity) from slope of the Load against Deflection diagram, we can also determine the modulus of elasticity from torsion test. By conducting torsion test of a material, we can determine its shear modulus (modulus of rigidity) and with the relationship of modulus of elasticity, modulus of rigidity and Poisson’s ratio, we can determine the modulus of elasticity of the given material. Other way of determining the modulus of elasticity is by drawing the stress – strain curve of the material. The slope of the curve of stress – strain diagram is the modulus of elasticity of the material.

10

5.0 CONCLUSION When a load is applied to a beam, either on a single point or distributed along the beam, deflection will occur on the beam. This deflection that occurs can be mathematically determined. Due to errors will conducting this experiment, we can assume that there will be different of result from theoretical values and experimental (practical) values. It is advisable that in every beam design, a margin of safety and thorough testing is needed before the design of the beam is used. It is also important to ensure the testing of the beam design match real world situation as close as possible.

11

6.0 REFERENCES 1.0 Ferdinand P. Beer, E. Russell Johnston Jr. , John T. DeWolf. "Mechanics of Materials". 2002. McGraw-Hill. New York. 2.0 Russell C. Hibbeler. “Mechanics of Materials”. 2007. Prentice Hall.

7.0 RAW DATA

12

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MR. SUHAIME SUPARMAN

GROUP: EMD4M12A MOHAMAD FIKRI BIN MOHD SHAHIDAN – 2010315603 MOHD SAFWAN BIN ABDULLAH – 2010176119 MOHAMMAD ILIYAS HAKIM BIN IBRAHIM – 2010967927

1

ABSTRACT When a beam is supported at two different point and loads are applied anywhere on the beam, we can mathematically determine the deflection that occurs on the beam. With the deflection know, we can also determine the Young’s Modulus (modulus of elasticity) of the material used in the experiment. The following procedures will explains how the experimental values of the deflection of beam and the modulus of elasticity of the beam are determined.

2

TABLE OF CONTENT 1.0 INTRODUCTION……………………………………………………………………………………………..4 1.1 GENERAL BACKGROUND…………………………………………………………………….4 1.2 THEORY………………………………………………………………………………………………4 2.0 EQUIPMENTS AND PROCEDURES……………………………………………………………………6 2.1 EQUIPMENTS………………………………………………………………………………………6 2.2 EXPERIMENTAL PROCEDURES…………………………………………………………….6 3.0 RESULTS………………………………………………………………………………………………………….7 3.1 TABLE OF RECORDED RESULT……………………………………………………………..7 3.2 GRAPH OF RESULT……………………………………………………………………………….8 3.3 CALCULATION OF RESULT……………………………………………………………………9 4.0 DISCUSSION…………………………………………………………………………………………………..10 5.0 CONCLUSION…………………………………………………………………………………………………11 6.0 REFERENCE…………………………………………………………………………………………………….12 7.0 RAW DATA………………………………………………………………………………………………………12

3

1.0 INTRODUCTION 1.1 GENERAL BACKGROUND If a beam is supported at two different point and loads is applied anywhere on the beam, a deflection of beam will occur. When the load are applied either on the outside of the inside of the support (in this case, the loads are applied on the outside of the support), the deflection of the beam can be mathematically determined by using the beam’s properties and geometries.

1.2 THEORY The figure below is the how the setup of the experiment will look like.

Where; R = Radius of curvature of beam, L = length of the beam, y = the deflection of the beam, x = the distance from the end of beam to support, W = weight of the load,

4

Pure Bending Theory;

(

)

( ⁄ ) ⁄

Therefore:

⁄ ⁄ and;

( ) ⁄

Therefore;

( )

(

) ( ) (

) ( )

Graph of W against y;

5

2.0 EQUIPMENTS AND PROCEDURES 2.1 EQUIPMENTS 1. 2. 3. 4. 5. 6. 7. 8. 9.

Weight Load holders Hanger Universal magnetic stand – to hold the gauge Gauge – to read the deflection Ruler – To measure the length of beam Screw driver Pliers Beam – (mild steel, aluminium, brass) – the beam should be rectangular, long and thin. The exact dimension of the beam should be measured and recorded 10. Vernier Caliper – To measure beam width

2.2 EXPERIMENTAL PROCEDURES 1. The dimension (width, height and length) of all the beams are measured with the vernier caliper and ruler and are recorded. 2. The 2 load holders are placed on the beam (mild steel) at a distances 15cm from each of support bars. 3. The beam is placed on the support bars. 4. The gauge is attached to the universal magnetic stand and is position at the middle of the beam. 5. The reading of the gauge is reset to zero. 6. Loads of 2N are added at each of the load holders simultaneously and an increment of 2N is added until a load of 16N is achieved at each load holders. 7. As the loads are added, the deflection reading from the gauge for each of the load is recorded. 8. Steps 2 to 7 are repeated for the remaining 2 beams (aluminium and brass). 9. The recorded deflections of the beams are then tabulated.

6

3.0 RESULTS Load (N)

Beam, Max Deflection (mm) Mild Steel

Aluminium

Brass

0

0

0

0

2

0.15

0.24

0.15

4

0.4

0.49

0.32

6

0.635

0.735

0.49

8

0.88

0.98

0.66

10

1.125

1.225

0.82

12

1.37

1.47

0.99

14

1.61

1.71

1.155

16

1.85

1.91

1.32

7

8

Calculation: Mild Steel: According to the graph drawn, the slope of the deflection for mild steel is 8431;

[

(

) ] ( )

( [

) (

) (

]

)

Aluminium: According to the graph drawn, the slope of the deflection for aluminium is 8272;

[

(

) ] ( )

( [

) (

) (

)

]

Brass: According to the graph drawn, the slope of the deflection for brass is 12040;

( [

) ] ( )

( [

) (

) (

)

]

9

4.0 DISCUSSION 1. Compare the value of E (modulus of elasticity) obtained from this method with their theoretical value. After conducting this experiment and calculate the data that was recorded, the value of E (modulus of elasticity) of Mild Steel is 220.19GPa, Aluminium is 68.93GPa and Brass is 100.33GPa. Whereas theoretical value of these beams (from text book) such as Mild Steel is 210GPa, Aluminium is 69GPa and Brass is 102GPa.

2. Comment on your result. The calculated E (modulus of elasticity) of the beams is not too different from their theoretical value. Such examples are the experimental value of E of Aluminium is 68.93GPa and its theoretical value is 69GPa. The different of this value is 0.07GPa or 0.10% different. Whereas the experimental value of E of Brass is 100.33GPa and its theoretical values is 102GPa. The different of value is 1.67GPa or 1.64%. Finally the experimental value of E for mild steel is 220.19GPa while its theoretical value is 210GPa. The different between these two values is 10.19GPa or 4.85%. The different in values between experimental and theoretical can be cause by error while conducting the experiment itself. In every experiment there will be error but it is the main goal to minimize these errors. Errors in values that are less that 5% are acceptable. 3. What are the other methods available to determine E of the materials. Other than determining the E (modulus of elasticity) from slope of the Load against Deflection diagram, we can also determine the modulus of elasticity from torsion test. By conducting torsion test of a material, we can determine its shear modulus (modulus of rigidity) and with the relationship of modulus of elasticity, modulus of rigidity and Poisson’s ratio, we can determine the modulus of elasticity of the given material. Other way of determining the modulus of elasticity is by drawing the stress – strain curve of the material. The slope of the curve of stress – strain diagram is the modulus of elasticity of the material.

10

5.0 CONCLUSION When a load is applied to a beam, either on a single point or distributed along the beam, deflection will occur on the beam. This deflection that occurs can be mathematically determined. Due to errors will conducting this experiment, we can assume that there will be different of result from theoretical values and experimental (practical) values. It is advisable that in every beam design, a margin of safety and thorough testing is needed before the design of the beam is used. It is also important to ensure the testing of the beam design match real world situation as close as possible.

11

6.0 REFERENCES 1.0 Ferdinand P. Beer, E. Russell Johnston Jr. , John T. DeWolf. "Mechanics of Materials". 2002. McGraw-Hill. New York. 2.0 Russell C. Hibbeler. “Mechanics of Materials”. 2007. Prentice Hall.

7.0 RAW DATA

12

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