Lecture 10 – Torsion

Instructor: Prof. Marcial Gonzalez

Fall, 2021ME 323 – Mechanics of Materials

Reading assignment: 4.1 – 4.5

Last modified: 8/16/21 9:15:07 AM

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Torsional deformation (@ ME 323)- Geometry of the solid body: straight, slender member with circular

cross section that is either constant or that changes slowly along the length of the member.

- Kinematic assumptions: the axis remains straight and inextensible. Cross sections, which are plane and are perpendicular to the axis before deformation, remain plane and perpendicular to the axis after deformation. Radial lines remain straight and radial as the cross section rotates about the axis.

- Material behavior: isotropic linear elastic material; small deformations.

- Equilibrium: the above assumptions reduce the problem to a one-dimensional problem!!

Torsion

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Torsional deformation- Geometry of the solid body: straight, slender member with circular

cross section that is either constant or that changes slowly along the length of the member.

Torsion

lug-wrench

stepped shaft

shaft-gear system

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Torsional deformation- Kinematic assumptions: the axis remains straight and inextensible.

Cross sections, which are plane and are perpendicular to the axis before deformation, remain plane and perpendicular to the axis after deformation. Radial lines (e.g., CD and FE) remain straight and radial as the cross section rotates about the axis.

Torsion

+Torque

+Angle of rotation

Experiment

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Torsional deformation- Kinematic assumptions: the axis remains straight and inextensible.

Cross sections, which are plane and are perpendicular to the axis before deformation, remain plane and perpendicular to the axis after deformation. Radial lines (e.g., CD and FE) remain straight and radial as the cross section rotates about the axis.

Torsion

Experiment

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Torsional deformation- Kinematic assumptions: the axis remains straight and inextensible.

Cross sections, which are plane and are perpendicular to the axis before deformation, remain plane and perpendicular to the axis after deformation. Radial lines remain straight and radial as the cross section rotates about the axis.

Torsion

Strain-displacement relationship

Radial position

Shear strain

Solid circular cylinder Tubular circular cylinder

Notice that the shear strain varies linearly with the distance from the axis ( )

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Torsional deformation- Material behavior: isotropic linear elastic material; small deformations.

Hooke’s law … (for homogeneous or uniform members)

Torsion

Radial position

Shear stress

Solid circular cylinder Tubular circular cylinder Central core of one materialbonded to an outer tubular sleeve

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Torsional deformation- Equilibrium: the above assumptions reduce the problem to

a one-dimensional problem+ Resultants for homogeneous materials (recall lecture 4)

Torsion

Polar moment of inertiaTorque-twist equation

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Torsional deformation- Geometry of the solid body: straight, slender member with circular cross

section that changes slowly along the length of the member.- Kinematic assumptions: the axis remains straight and inextensible. Cross

sections, which are plane and are perpendicular to the axis before deformation, remain plane and perpendicular after deformation. Radial lines remain straight and radial as the cross section rotates about the axis

- Material behavior: isotropic linear elastic material; small deformations.

- Equilibrium: (torque-twist equation)

Torsion

Shear strain

Total angleof rotation

Homogeneous:

Homogeneous:

Homogeneous, constant cross section: Q: units???

�max = r0�

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⌧max = Gr0�

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Example 14 (review):- Determine the internal torque in each member.- Determine the rotation angles at A and B.- Determine the state of stress at four differentpoints in each member.

Torsion

�B =L2

G2Ip2(TA + TB)

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Example 15:Determine the maximum shear stress in the steel and the maximum shear stress in the aluminum.

Torsion

1) Free body diagram2) Equilibrium equations3) Torque-twist behavior4) Compatibility conditions5) Solve for unknowns

staticallyindeterminate

structures

Any questions?

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Torsion