To investigate the relation between Shear Stress and Shear Strain and to Determine the Modulus of Elasticity of the steel wire.nt 1 PDF

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This is a Lab Experiment performed using Torsion Apparatus....

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Experiment – 1 1.1 – Objective: The Objectives of the following assignment are  To investigate the relation between Shear Stress and Shear Strain  Determine the Modulus of Elasticity of the steel wire.

Figure – 1: Modulus of Elasticity Apparatus Source: www.google.com/images

1.2 – Apparatus:      

Torsion and Shaft apparatus Hangers Weights Meter Rod Micrometer Vernier Calipers 1

1.3 – Explanation of Torsion Apparatus: Torsion of shaft apparatus includes a shaft of circular section, two measuring scales and a pulley with a frame. The main purpose of the pulley with hanger is to apply some load on the circular shaft. Similarly, the scales attached to the frame are used to measure the torsion in the circular shaft. Actually, two scales are used, one at the front and one at the back. The measuring arms (scales) are used to measure the magnitude of the torsion at the front and the back of the circular shaft respectively. The front is the portion of the shaft that is near to the pulley and the back is the portion of the shaft near the back support of the frame. The main purpose of the frame is to support the shaft and balance the apparatus on the surface.

1.4 – Theory: The related concepts to this experiment are explained in the theory below. These concepts are necessary in fully understanding the working of the apparatus as well as the working done in achieving the objectives.

1.4.1 – Stress and Tension: Stress is defined as force per unit area for any material. Force is due to the reaction forces inside the material. The maximum stress a material can withstand is called the breaking stress or ultimate tensile stress (UTS). Tension means that the material is under the action of some force, it generally refers to contraction or elongation.

1.4.1.1 – Types of Stress: There are generally three types of stresses:  Tensile stress (or tension) is the force per unit area applied on a material that causes the material’s length to change that is it contracts or expands depending upon the direction of the stress. Volume generally stays constant.

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Figure – 2: Tensile Stress Source: www.google.com/images

 A shear stress is the stress that acts parallel to the plane of the body. It arises due to the force component parallel to the cross-section of the body.

Figure – 3: Shear Stress Source: www.google.com/images

 Normal stress is a stress caused by an axial force. It will occur when a body or face is placed in tension or compression.

Figure – 4: Normal Stress Source: www.google.com/images

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1.4.2 – Strain: Strain is the change per unit of a body by any applied stress. This change can be in the Length, Shape, and Volume etc. depending upon the type of stress that produces the strain.

1.4.2.1 – Types of Strain: There are generally three types of Strain  The Strain produced in length due to tensile stress is called tensile strain.  The Strain produced in volume due to Normal stress is called Normal strain.  The Strain produced in shape due to Shear stress is called Shear strain

1.5 – Elastic Limit: Elastic limit is the limit of stress up to which when it is applied the body is not deformed permanently. When the force within the elastic limit is removed the body takes back its original shape. Elastic limit is an experimentally determined quantity found by the Stress-Strain Graph of a material.

1.6 – Plastic Limit: Stresses exceeding the elastic limit cause the materials to deform plastically. Hence elastic limit marks the start of plastic behavior. Materials are classified depending upon their plastic range. Ductile materials have a high range of plasticity while brittle material fracture soon after the elastic limit is crossed.

1.7 – Proportional Limit: Proportional Limit marks the end of linearly proportional behavior between the Stress and Strain for a material. It is also an experimentally determined limit. For most materials the elastic and the proportional limit overlap hence it is difficult to recognize while in other materials the elastic limit may stretch well above the proportional limit.

1.8 – Hooke’s Law: Hooke’s law states that “Stress applied on a body is linearly proportional to the strain provided the stress doesn’t exceed the proportional limit for the material.”

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Hooke’s law is related to the proportional limit unlike the wide misconception that Hooke’s law depends upon the elastic limit. This is because for much material the elastic limit ends at the proportional limit of the material.

1.9 – Ratios of Stress and Strain: The ratios of stress and strain can give three types of constants depending on the type of stress and strain explained earlier:    

Young's Modulus E is the ratio of tensile stress to tensile strain. The Bulk Modulus K is the ratio of Normal stress to Volumetric Strain. The Shear Modulus G is the ratio of axial stress to angular displacement. Poisson's ratio tells us about responses in the directions orthogonal to the uni-axial stress.

1.10 – Young’s Modulus: Young's modulus is the basic property of a material to be measure of its resistance towards tensile stress. It is named after the 19th – century British scientist Thomas Young. The concept was further elaborated in 1727 by Leonhard Euler, and the Italian scientist Giordano Riccati first performed the experiments relating to or using Young’s Modulus.

1.11 – Modulus of Rigidity: Modulus of Rigidity is the ratio of Shear stress to angular displacement. Shear stress is the stress when force is applied axially to one of the faces of the body while opposing face experiences the same force in the opposite direction due friction etc. Modulus of Rigidity can be determined experimentally by the slope of a stress-strain curve formed during tensile testing on a sample of material.

1.12 – Dependence on Direction: Young's modulus is not bound to be same in all directions i.e. it is not anisotropic. Metals and ceramics are isotropic, and their Young’s modulus and other mechanical properties are the same in all directions. However, they can be doped with certain impurities, and metals can be worked mechanically improve their grain structures to increase hardness and directional mechanical properties. These materials then become anisotropic. Anisotropic behavior can be seen in many composites as well. Other common materials include wood and reinforced concrete. These direction properties are not always a flaw and can be used by engineers for practical purposes.

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1.13 – Shear Modulus of Metals The shear modulus of metals usually decreases with increasing temperature. At high pressures, the shear modulus increases with the applied pressure. Correlations between the melting temperature, vacancy formation energy, and the shear modulus have been observed in many metals. Several models exist that attempt to predict the shear modulus of metals (and possibly that of alloys). Shear modulus models that have been used in plastic flow computations include: 1. The MTS shear modulus model developed by and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress model. 2. The Steinberg-Cochran-Guinan (SCG) shear modulus model developed by and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model. 3. The Nadal and LePoac (NP) shear modulus model that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.

1.14 – Procedure: 1. Apply the initial weights on the hanger to clear off the ripples in the wire.. 2. With the hanger in position apply a load to the hanger and read the vertical displacement of the loading plate relative to the initial value using the scale provided on the apparatus. 3. The reading for the first load should be neglected as it’s only to remove the initial wrinkles. 4. Check all the apparatus for the Zero error. 5. Repeat the experiment for increasing load and record the vertical displacement of the loading plate in each case. 6. Unload and note the corresponding reading with the load decreasing. 7. Take the mean of readings. 8. Calculate the “Modulus of Elasticity” of the material.

1.15 – Observations and Calculations in Inches:        

Least count of scale of apparatus = 0.5cm (0.1968inch) Least count of micrometer = 0.01mm Least count of meter rod = 1mm Length of wire = 88cm (34.64inch) Diameter of wire =0.86mm (0.05354inch) Initial load =5lb Area of wire =πd2/4=8.99×10-4 inch2 L/A=3.85×104 inch-1

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1.16 – Readings: No of obs

Load-p (lbs)

Extension del l INCH loading

unloading

average

Load/extension

Modulus of Elasticity M psi

1

0

0

0

0

0

0

2

10

0.019

0.019

0.019

526

20.25

3

15

0.03937

0.03937

0.03937

384

15

4

20

0.0590

0.0590

0.0590

369

14.24

5

25

0.0787

0.0787

0.0787

320

12.32

6

30

0.098

0.098

0.098

306

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The average Modulus of Elasticity for Steel wire comes out to be 14.762Mpsi. The real value is 29Mpsi so there is 50.90345% error.

1.17 – Graph

Load-Extension Graph 600 500

Strain

400 300 200 100 0 0

5

10

15

20

25

30

35

Stress

1.18 – Theoretical Values: 7

The theoretical values of the Scotch Yoke mechanism can also be found by the formulas that are derived from its mechanism. The values of velocity and that of acceleration depend upon the crank position. They are also the ideal values of a graph.

1.19 – Experimental Values: The values that get due to our readings are the experimental values. They may differ from the theoretical values due to certain reasons

1.20 – Reasons of Deviation: The experimental and theoretical values may differ due to the following reasons:     

We neglect the value of angular velocity while it may not be same in every stroke. There may be random error due to lack of experience of the worker There may be fault in the apparatus or limitation in it due to its least count. There can be friction between the moving parts There is a least count and so a limitation for accurate reading of the scale

1.21 – References: www.google.com/images http://www.strucalc.com/normal-stress-bending-stress-shear-stress/ https://en.wikipedia.org/wiki/Shear_stress https://simple.wikipedia.org/wiki/Stress_(mechanics) https://www.engineeringtoolbox.com/modulus-rigidity-d_946.html https://en.wikipedia.org/wiki/Shear_modulus

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