PROJECT REPORT
Mechanics of Material-1
Lab
Submitted To:
Ma’am Anam Rehman
Submitted By:
2016-ME-76 Fahad Ali
2016-ME-83 Shaharyar Saleem
2016-ME-87 Ahsan Razzaq
2016-ME-93 Saad Tanveer
2016-R/2015-ME-95 Ali Hayajneh
Department Of Mechanical Engineering
University Of Engineering and Technology, Lahore
TABLE OF CONTENTS
Objectives ……………………………………………… 3
Related Theory ………………………………………… 3
2.1 Shaft …………………………………………………………… 3
2.2 Types of Shafts ……………………………………………….. 4
2.3 Torque ………………………………………………………… 5
2.4 Torsion ………………………………………………………… 5
2.5 Moment of Inertia …………………………………………….. 6
2.6 Polar Moment of Inertia ……………………………………… 6
2.7 Shear Stress …………………………………………………… 7
2.8 Shear Strain …………………………………………………… 7
2.9 Modulus of Rigidity (G) ……………………………………… 7
3. Experiment performed on Fabricated Apparatus…… 8
4. Fabricated Apparatus under Observation…………… 9
5. Laboratory Manuals of different universities……….. 10
5.1 University of Lahore, Dept. of Mechanical Engineering……… 10
5.2 University of Memphis, Dept. of Mechanical Engineering…… 14
6. Different Materials under consideration…………….. 17
6.1 Brass Specimen……………………………………………….. 17
6.2 Steel Specimen……………………………………………….. 18
6.3 Aluminum Specimen…………………………………………. 19
7. Discussion and Conclusion……………………………. 20
8. References………………………………………………. 21
8.1 Books…………………………………………………………. 21
8.2 Articles………………………………………………………… 21
FABRICATION OF COPPER SHAFT APPARATUS
1. Objectives:
To learn about the torsion theory of different shafts made up of different materials.
To get familiarize and impose the principle of torsion testing.
To calculate shear stress, shear strain and modulus of rigidity of the copper shaft.
To find the relationship between angle of twist and applied torque for the material under observation.
To select a material most suitable, both economical and performance wise, in the production of different engineering components.
Related theory:
2.1 Shaft: Shaft is a mechanical device whose primary function is to transmit mechanical power from one part to another/from one machine to another in a mechanical device.
Figure shows a simple demonstration of a circular shaft
There are multiple common types of shafts, two of which are discussed below.
2.2 Types of shafts
Transmission shafts.
Machine shafts
Transmission shafts: It works as power transmitting device between source and power absorbing machine.
Figure shows high strength transmission shafts
Machine shafts: It is considered as necessary part of a machine which is used to transmit power within the machine.
Figure shows Turbine Generator Shaft
Standard sizes:
For machine shafts, the commonly used size is up to 25mm steps of 0.5mm while for the Transmission shafts, there are several sizes mentioned below:
25mm to 60mm with 5mm steps
60mm to 110mm with 10mm steps
2.3 Torque: Torque is described as twist to an object. Mathematically it is defined as vector cross product of position vector and force vector.
Units:
Dimensionally, torque equals Force Times Distance. So symbolically, torque has dimensions L2MT-2. Thus its units are Newton Metre (N.m) or Joule per Radian.
2.4 Torsion: When a solid object is subjected to a torque, twisting is produced about its longitudinal axis. This process of twisting is called torsion. The basic units in which Torsion is measured are Pa and psi.
2.5 Moment of Inertia: Generally, it is defined as body’s tendency to resist angular acceleration. In engineering we discuss it as the ratio of the net angular momentum of a system to its angular velocity around a principal axis.
2.6 Polar Moment of Inertia: It is defined as shafts opposition to the angular distortion when torque is applied onto the shaft. It is also known as second polar moment of area.
2.7 Shear Stress: The resultant force which tends to deform a material by slippage along a plane or planes parallel to the imposed stress. This deformation is called shear stress
2.8 Shear Strain: It is the ratio of change in length produced due to shear force and original length of the specimen. This ratio is equal to the tan φ (where φ is the angle opposite to the deformed length and adjacent to the original length)
2.9 Modulus of Rigidity (G): It is also known as shear modulus. It is the ratio of shear stress to shear strain. Experimentally modulus of rigidity is calculated with the help of stress-strain curve. The slope of stress-strain curve gives us value of modulus of rigidity. Mostly materials have large shear modulus; therefore we use base units MPa/GPa to describe it.
stress ∝ strain
3. Experiment Performed on Fabricated Apparatus
Objective:
To determine the modulus of rigidity for Copper material of circular shaft.
Apparatus:
Screw Gauge, Meter Rod, Vernier Calliper, Weights and Hangers.
Torsion of Copper shaft apparatus; It includes a pulley with a frame, two measuring rulers and a circular shaft made of Copper. The main aim of pulley with support is to put on some weight on the circular shaft. Correspondingly, to measure the amount of twist in the circular shaft, two scales are used, one at the front and one behind.
To measure the amount of twisting at the both ends of the circular copper shaft, the measuring scales are used. The frame is used to provide support to the circular shaft. It is also used to balance the apparatus on the surface.
Procedure:
To perform this experiment and calculate the rigidity:
Place the apparatus of copper specimen on a smooth horizontal surface.
Using meter rod, measure the effect length of the copper shaft.
Using Screw Gauge, measure the diameter of the shaft.
Amend the Zeros at 1st and 2nd measuring scales.
Place a weight of 1 lb in the hanger.
Measure the 1st and 2nd angle of twist in radians.
Now with the increment of 1 lb Weight each time, take 6 loading values and then 6 unloading values.
Calculate the modulus of rigidity of the Copper shaft using the formula:
G= TL/Jθ
Calculations:
Effective Length of shaft (L) = ___________ in.
Diameter of shaft (D) = ___________ in.
Diameter of torque pulley (D) = ___________ in.
Radius of Torque Pulley (R = D/2) = ___________ in.
Polar Moment of inertia of shaft = ___________ in.
(J = πd^4/32)
No. of Obs.
Load
W
(lbs) Torque
WR
(lb-in) Angle of twist at 1st measuring arm
θ_1
(rad) Angle of twist at 2nd measuring arm
θ_2
(rad) Angle of twist for effective length
θ= θ_1-〖θ2〗_
(rad) Modulus of Rigidity
G = TL/J θ
(psi)
Loading Unloading Average Loading Unloading Average
1.
2.
3.
4.
5.
TABLE
4. Fabricated Apparatus under Observation:
Torsion of shaft Apparatus is used to find the amount of torsion in any shaft made up of specific material.
Fabrication:
The apparatus is fabricated by taking the following steps:
The apparatus used is portable; i.e. the different parts of the apparatus can be separated from each other and can be transported in separate pieces easily.
Nuts and bolts are used to join the different parts of the apparatus. This allows us to use the shaft of any material for performing the experiment. Hence, we can take the different materials under consideration easily.
Extra bearings are used which allows smooth rotation.
Figure shows the apparatus during fabrication process.
5. Lab Manuals of Different Universities
5.1 UNVERSITY OF LAHORE, DEPARTMENT OF MECHANICAL ENGINEERING
Objective:
To determine the shear modulus and shear stress for a given shaft using torsion apparatus.
Working Principle (Purpose):
A torsion test is quite instrumental in determining the value of modulus of rigidity (ratio of shear stress to shear strain) of a metallic specimen. Member in torsion are encountered in many engineering applications. The most common application is provided by transmission shafts, which are used to transmit power from one point to another. For example, the shaft used to transmit power from the engine to the rear wheels of an automobile. These shafts can either be solid or hollow.
Figure shows torsion of shaft apparatus (currently present in Department of Mechanical Engineering, Mechanics of Materials Lab, University of Lahore)
Theory:
Torsion is the engineering word used to describe the process of twisting a member about its longitudinal axis. A torsion test is quite instrumental in determining the value of modulus of rigidity (ration of shear stress to shear strain) of a metallic specimen.
Member in torsion are encountered in many engineering applications. The most common application is provided by transmission shafts, which are used to transmit power from one point to another. For example, the shaft used to transmit power from the engine to the rear wheels of an automobile. These shafts can either be solid or hollow.
Considering a cylindrical bar with one end being twisted, the twisting torque T is resisted by the shear stress τ existing across the specimen section. This shear stress is zero at the center of the bar, increases linearly with its radius and finally reaches its maximum value at the peripheral of the bar. If the cylindrical bar with a length of L, the twisting moment can be related to the shear stress as follow:
T/J= Gϕ/l= τ/ρ
Where;
J is the polar moment of inertia, mm2
G is the shear modulus, N/mm2
ϕ is degree of rotation, radian
ρ is the radius of the cylindrical bar, mm or in
l is the length of the cylindrical bar, mm or in
τ is the shear stress, N/mm2
Within the elastic range of deformation, the shear stress can be calculated according to the equation:
τ= Tρ/J
For a solid cylindrical specimen, the polar moment of inertia is:
J= (πD^4)/32
Therefore, the shear stress can be determined by:
τ= (T D/2)/((πD^4)/32) = 16T/(πD^3 )
Using equation 1: G= Tl/ρϕ
Putting the value from equation 2 and J for circular shaft, we get:
G= Tl/Jϕ
As 3.14 rad = 180o then 1 rad = 57.3o
So, G=57.3 Tl/Jϕ
Procedure:
Set the clamping distance of the supports at 600 mm and clamp the test shaft at both ends.
Position the lever and set the dial gauge tip on the lever groove in compressed form.
Set the gauge scale to Zero.
Note that 1 mm reading of the dial gauge measures 1o angle of twist.
Now put the load in suitable increments and for each load record the angle of twist.
Calculate the corresponding values of shear modulus and shear stress.
Repeat the experiment with a new span of 200 mm of the same shaft.
You can also repeat the entire experiment with new test shaft specimen.
Observations and Calculations:
Diameter of shaft = d = 9.1 mm
Span length = L = 500 mm
Moment arm = R = 60 mm
Applied load = W
Polar moment of inertia = J = (πD^4)/32 = 672.9 mm4
Torque = T = F x R
Angle of twist = ϕ
Shear stress = τ= 16T/(πD^3 )
Shear modulus of rigidity = G = 57.3 x TL/Jϕ
TABLE
Shaft (d = 9.1 mm, J = 672.9 mm4)
Obs. M W R T ϕ τ G
(kg) (N) (mm) (N.mm) (o) (N/mm2) (N/mm2)
1. 2.04×10-4 2 80 160 0.18 1.08 37846.14
2. 4.07×10-4 4 80 320 0.33 2.16 41286.4
3. 6.12×10-4 6 80 480 0.55 3.25 37158.03
4. 8.15×10-4 8 80 640 0.75 4.32 36322.3
5. 1.01×10-3 10 80 800 0.9 5.41 37846.14
5.2 UNIVERSITY OF MEMPHIS, DEPARTMENT OF MECHANICAL ENGINEERING
Objective:
The objective of this experiment was to determine the relationship between angle of twist and applied torque for a rod as well as the relationship between the deflection of length of the rod at the specified torque.
Theory:
Torque is defined as a moment that acts about a member’s longitudinal axis. A member that has had torque applied to it such that it deforms along its longitudinal axis is said to be under torsion. The torsion presents itself as shear strain which is equal to the angle of twist along the longitudinal axis. Because this experiment uses cylindrical specimens, the theory discussed will pertain only to members of circular cross-section.
As a member of circular cross-section is twisted along its longitudinal axis, the cross-sectional area remains constant without deforming. Because there is no deformation in the plane of cross-section, it is implied that there is no strain in the member’s latitudinal direction. Therefore, the components of shear stress act only in the radial direction. The shear strain of a member is the product of a shear stress. Shear stress occurs in cylindrical member when torque is applied. The shear stress is a function of the radius of the circular cross section. Along the longitudinal axis, the shear stress is null. At any given point along the radius of cross-section, the shear stress is a function of radius ρ, denoted by τ_ρ where J is the polar moment of inertia. The further out from the axis, a measurement is taken for ρ, the larger the calculated shear stress will be. At the circumference of the member, the shear stress will be at its greatest value, τ_max.
τ_ρ= T_ρ/J (Equation 1)
To calculate polar moment of inertia J, for a circular shaft, employ the following equation where D is the diameter of the cylindrical member.
J=π/32 D^4 (Equation 2)
If the shear stress induced in the member is below the proportional limit of the material, then Hooke’s Law may be applied so as to calculate the material’s modulus of rigidity. In other words, if the stress causes only elastic or non-permanent deformation, the materials torsional stiffness can be determined. To apply Hooke’s Law, the shear stress is related to shear strain by:
τ=Gγ
Where G is the modulus of rigidity and γ is the shear strain.
γ= ρ dϕ/dx
Above equation implies that the shear strain is proportional to the product of the radius of the member and the change in the angle of twist with respect to the longitudinal axis. Solving the above equations yields the final equation as:
T/GJ= dϕ/dx
Separating the above equation and integrating with respect to the longitudinal axis will yield an expression that describes the angle of twist for a member that is prismatic and experiencing a constant internal torque.
ϕ= TL/GJ
Equipment:
TQ Hi-Tech Torsion Experiment Apparatus (SN: HFC.2)
Copper Rod
Weights in 2 lb increments up to 12 lbs.
Procedure:
Part I:
Place the torsion experiment apparatus on a hard, flat surface with the pulley and load hanger over the edge of the table.
Load a cylindrical specimen into the device by tightening the rod into the chuck and clamp at either end of the apparatus.
Wind the cord of the load hanger in a clockwise fashion so as to ensure the applied load on the hanger will induce a torque.
Position the two angle scales of the torsion experiment apparatus at an arbitrary distance from one another on the rod. It is easier to use a round number such as 8 or 10 inches, record this value as it will be used as the length L needed to calculate the modulus of rigidity of the material.
Calibrate the pointers of the angle meters by placing the needles at zero.
Add two pounds of weight to the load hanger suspended from the pulley. Ensure that the cord is still wrapped clockwise around the pulley. Record the indicated angles on both of the angle scales, as this is the angle of twist.
Increase the load of the hanger by two pound and again record the indicated angles on both angle indicators.
Continue to add two pound increments to the load hanger and record the indicated angles until a total weight of 12 pounds has been achieved. Repeat step 5 through 8 for two to four trials. Take the average of the angles used.
Part II:
Position the load hanger ensuring the cord is wrapped in clockwise direction around the pulley. Zero the angle indicators.
Place 10 pounds of weight on the hanger.
Record the position of the angle indicators and the indicated angles on each.
Reposition the angle scales in increments of two inches up to twelve inches apart while maintaining a load of ten pounds.
Repeat steps 2 through 5 for a desired number of trials.
Calculations:
Length of Twist = ________________ in.
Polar Moment of Inertia = ________________ in4
Diameter of Pulley = ________________ in.
Diameter of Test Rod = ________________ in.
No. of Obs. Weights
(lbs) Twist at A
(in degrees) Twist at B
(in degrees) ∆AB
(in radians) Applied Torque
(lbf*in)
1
2
3
4
5
6. Different Materials under Consideration:
6.1 Brass Specimen
Initial measurements of Brass rod and pulley are given below following the measurements of weight induced torsion and calculated modulus of rigidity for cylindrical specimen of Brass.
Length of Twist 10 in.
Polar Moment of Inertia 1.51×10-4 in4
Diameter of Pulley 3 in.
Diameter of Test Rod 0.198 in.
Weights
(lbs) Twist at A
(in degrees) Twist at B
(in degrees) ∆AB
(in radians) Applied Torque
(lbf*in)
2 5 3.5 0.02618 3
4 10 5.5 0.07854 6
6 14 7.5 0.11345 9
8 18.25 9.5 0.15272 12
10 22.5 11.25 0.19635 15
12 27.5 14 0.23562 18
Graph of angle of twist vs. applied torque of a cylindrical brass specimen which contains a trend line which will be used to calculate the experimental modulus of rigidity
6.2 Steel Specimen
Initial measurements of Steel rod and pulley are given below following the measurements of weight induced torsion and calculated modulus of rigidity for cylindrical specimen of Steel.
Length of Twist (in.) 10
Polar Moment of Inertia (in4) 3.96×10-4
Diameter of Pulley (in) 3
Diameter of Test Rod (in) 0.252
Weight
(lbs) Twist at A
(in degrees) Twist at B
(in degrees) ∆AB
(in radians) Applied Torque
(lbf*in)
2 1 0.25 0.01309 3
4 1.75 0.75 0.01745 6
6 2 0.75 0.02182 9
8 3 1.25 0.03054 12
10 3.5 1.5 0.03491 15
12 4 1.75 0.03927 18
Graph of angle of twist vs. applied torque of a cylindrical steel specimen which contains a trend line which will be used to calculate the experimental modulus of rigidity
6.3 Aluminum Specimen
Initial measurements of Aluminum rod and pulley are given below following the measurements of weight induced torsion and calculated modulus of rigidity for cylindrical specimen of Aluminum.
Length of Twist (in.) 10
Polar Moment of Inertia (in4) 1.51×10-4
Diameter of Pulley (in) 3
Diameter of Test Rod (in) 0.198
Weight
(lbs) Twist at A
(degrees) Twist at B
(degrees) ∆AB
(in radians) Applied Torque
(lbf*in)
2 3.25 1.5 0.03054 3
4 6.75 3.25 0.06109 6
6 10.5 5 0.09599 9
8 14.25 6.75 0.13090 12
10 18 8.75 0.16144 15
12 21 10.25 0.18762 18
Graph of angle of twist vs. applied torque of a cylindrical aluminum specimen which contains a trend line which will be used to calculate the experimental modulus of rigidity.
7. Discussion & Conclusion:
When a given torque is applied to a material of any specimen, it’s modulus of rigidity tells us how much of the twist or rotation takes place. Higher value of modulus of rigidity results in a lesser amount of torsion. On the other hand, lower value of modulus of rigidity results in a greater amount of torsion for the same specified torque.
After performing the experiment using the aluminium specimen as the material, it was distinguished that it’s value of rigidity G was inconsistent with its value that was theoretically calculated. A calculation was performed including percent error for the different materials of specimens.
Material Calculated Modulus of Rigidity (lbf/in2) Theoretical Modulus of Rigidity (lbf/in2) % Error
Brass 5.12×1006 5.50×1006 6.91%
Steel 1.05×1007 1.15×1007 8.70%
Aluminium 6.24×1006 3.80×1006 64.21%
As seen above, the percent error for Brass and Steel specimen is below 10% which is acceptable but for the Aluminum specimen, the percent error was found to be more than 60% which is way off the acceptable range.
There are many explanations for my the theoretical and calculated values of the aluminum specimen are found to be inconsistent. A likely error can be that the material of the specimen tested was not Aluminum. If we don’t analyze thoroughly the material of specimen, we cannot state by being sure that which material is used while performing the experiment. Another possible explanation is that the specimen’s test rod was not of a constant radius. When the radius varies, the calculations would be different because of different value of J and hence it gives different values of G.
Other explanation is done by considering the indicated angles which are only limited to the nearest half degree value only. This restriction does not allow the twist angle of the material of specimen undergoing torsion experiment to be precise.
8. References
8.1 Books:
Seaburg, Paul; Carter, Charles (1997) Torsional Analysis of Structural Steel Members. American Institution of Steel Construction pt. 3
Case and Chilver “Strength of Materials and Structures”
Khurmi R S, (2014) “A text book of machine design”. New Delhi
Mechanics of Materials, 2nd Edition Timothy A. Philpot (2011)
Fundamentals of Material Science and Engineering; An Integrated Approach W.D. Callister, Jr and D.G. Rethwish (2008)
8.2 Articles:
https://en.wikipedia.org/
https://www.scribd.com/
https://www.green-mechanic.com/