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Descripción completa
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Descripción: Torsion
Trabajo teorico de torsion.Descripción completa
1
Job # 3: To perform the Torsion Test on:
Mild Steel Specimen
Cast iron Specimen
Purpose:
To study the shear stress and shear strain behavior of the two different materials. To study the failure patterns of these materials in torsion. To determine the mechanical properties of the samples like modulus of rigidity, shear strength, shear strain and ductility d uctility in torsion.
Apparatus:
10 Ton Buckton Universal Testing Machine
Vernier Callipers
Steel Rulers
Related Theory: Torque:
The twisting effect of a force or couple is called c alled Torque. Its unit are N-m. It is denoted by Double Headed arrows. Torsion:
It is the angular deformation produced by the torque about the longitudinal axis. Following figure shows the torque applied to circular shaft & its behavior.
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Difference between Torque and Bending Moment:
Major difference is the plane of application of load or force. Torque is produced by the force which is applied parallel to the cross sectional axis at an eccentricity while moment is produced by the force which is applied along any cross sectional axis without any eccentricity. But basically both are moments. Fig 1. Clearly shows the difference between the torque and the moment.
L
P
P
L
P
e
P
Fig 1: showing difference between bending moment and torque
Bending Theory:
Assumptions in Bending Theory:
Material is homogenous
Material is isotropic
It is in elastic range i-e obeys the Hooks law
Plane section remains plane before and after bending.
Torsion Theory:
τ = Torsional shear stress (MPa) r = Radius of shaft or bar
T = Applied torque J = Polar moment of inertia (mm4) (torsional resistance of any cross section) G = Modulus of rigidity (modulus of elasticity in shear) θ =Angle of twist (rad) L = Length of sample
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Assumptions in Torsional Theory:
Material is homogenous
Material is isotropic
It obeys the Hooks law
Cross section of the material is circular
Plane section remains plane before and after bending
Length and radius of the sample are same for small angle of twist.
Stresses developed Under Torque:
Shearing stresses are developed in the bar when the bar is subjected to the torque. Consider a shaft subjected to the torque as is shown in figure below.
If we pass a cutting plane at point C, we will see that how forces due to the applied torque T are developed in the cross section which are parallel to the cross section. These internal shearing forces are resisted by polar moment of inertia.
Significance of Circular bar under torsion as compared to other sections:
An important property of the circular section is that when the circular section is subjected to torsion every cross section remains plane and undistorted. In other words we can say that various cross sections along the bar rotate through different amounts and each cross section rotate as a solid rigid slab. This property is only found in circular bars. But when a bar of square section is subjected to torsion its various cross sections wrap and do not remain plane.
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Fig: showing the difference in behavior of the circular and non-circular shaft
Shearing stress Distribution for torsion:
Shearing stresses produced due to torsion are maximum at the extreme outer fiber and zero at the neutral axis. While the direct/flexural shear stresses are maximum at the neutral axis and zero at the extreme outer fiber.
Put R = D/2
τmax
τmax
Note: this formula is for solid circular section
τmax
τmax
Fig: showing shear stress distribution for hollow circular and circular bar under torsion
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Polar Moment of inertia "j":
Geometric rigidity of the cross section is termed as polar moment of inertia or nd polar 2 moment of area. It is the resistance of a cross section against the nd twisting. It is equal to the sum of 2 moment of inertia about both cross sectional axis. Stiffness is the resistance of the material against bending while rigidity is the resistance of the materials against torsional loading.
For hollow circular sections polar moment of inertia is given by:
For circular sections polar moment of inertia is given by:
Torsional rigidity or Modulus of Rigidity "G":
When the material is subjected to pure twist loading the slope of the shear stress versus shear strain curve is termed as modulus of rigidity or modulus of elasticity in shear. It is denoted by "G".
τ
б
E
G
Є
γ
Fig: showing modulus of elasticity in tension test and modulus of rigidity in torsion test
The relationship between modulus of rigidity and modulus of elasticity is given as:
Where" is the Poisson's ratio
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Poisson's ratio:
The ratio of the lateral strain to longitudinal strain when the material is subjected to axial loading. And it is always less than 1.
For isotropic material Poisson's ratio is 0.25 - 0.4 For steel 0.3-0.35 (but 0.3 is mostly used) For concrete it is 0.15
Relation between yield strength in tension to that in torsion for mild steel:
√
This criteria is called VON MISES yield criteria.
Torsional Strength:
It is the maximum torsional stress that a material sustains before rupture. It is similar to tensile strength in tension test. Torsional Deformation:
It is the angular displacement of specimen caused by the specified torque in torsion. It is equal to angle of twist in radians divided by gage length or effective length. Torsional deformation =
Its units are rad/mm. Torsional Strain ():
It is the strain corresponding to specified torque. Torsional strain = torsional deformation x radius of the bar
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Where "Le" is the effective length or clear length between the jaws. Torsional Stress:
Shear stress developed in a material subjected to a specified torque in torsion test. For circular shaft
Expected Failure Modes of Mild Steel and Cast iron: Mild Steel:
Direction of application of load is at 45 degrees from horizontal axis. And hence shearing stress are maximum at 45 degrees from the direction of application of load i-e at the horizontal axis. And also mild steel is weak in shear so failure will be in horizontal plane.
Fig: showing the expected failure pattern of mild steel in torsion test
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Cast iron:
Maximum tensile stresses will develop at a plane 90 degrees from the line of action of the force and hence the cast iron is weak in tension so failure will be at 45 degrees from the horizontal axis. As twisting is a helical type of phenomenon and keep on changing its direction so the failing plane is also helical.
Fig: showing the failure pattern of cast iron i n torsion test
Note:
Shearing stresses are maximum on a plane at an angle of 45 degrees to the direction of application of the force and tensile stresses are maximum on a plane at an angle of 90 degrees to the direction of application of force.
Procedure:
Measure the total length, effective length and diameter of the sample.
Place the specimen in 10 Ton Buckton Universal testing machine.
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Rotate the handle of the machine counter clockwise and measure the degrees. (Note: 1 complete rotation of the handle is 4 degrees ro tation of the specimen). Repeat the same procedure until failure of the specimen occurs.
Observations & Calculations: Liver arm = la = 50.8 mm Diameter of the specimen = D = 17.77 mm Effective length = le = 139 mm
Comparison Graph Between Mild & Cast Iron in Torsion Test 500 450 400
) a P350 M ( 300 s s e r t 250 S g n 200 i r a e h150 S 100 50 0 0
0
0
0
0
1
1
Shearing Strain (Radians)
Comments:
Mild steel specimens failed at an angle of 90 degrees from the line of action of force showing the tensile failure as the cast iron is weak in tension. And mild steel specimen failed at an angle of 45 degrees from the line of action of force showing the shear failure of the specimen as the mild steel is weak in shear. From the graph Shearing strain produced in mild steel is much higher than the cast iron this shows the ductile behavior of the mild steel specimen.
