lecture 10 –torsionweb.ics.purdue.edu/~gonza226/me323/lecture-10.pdf · 2020. 8. 23. · torsion...
TRANSCRIPT
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
2
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
3
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
4
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
5
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
6
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 ( )
7
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
8
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
9
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�
L<latexit sha1_base64="hKyOt1VWGjAOsNnyqF7MOcJRNx4=">AAACEXicbVA9S8RAEJ34bfw6tbRZPISrjsTGawTRxsJCwVPhcoTJ3uZucTcJuxvxCPkLNv4VGwtFbG3Ezsq/4t5HoacPBh7vzTAzL8oE18bzPp2p6ZnZufmFRXdpeWV1rbK+caHTXFHWpKlI1VWEmgmesKbhRrCrTDGUkWCX0fXRwL+8YUrzNDk3/Yy1JXYTHnOKxkphpRZ0UUoMA4mmp2Qh8bZ0910VeiSIFdIiyHq8LE7KsFL16t4Q5C/xx6R60Ph6JwBwGlY+gk5Kc8kSQwVq3fK9zLQLVIZTwUo3yDXLkF5jl7UsTVAy3S6GH5VkxyodEqfKVmLIUP05UaDUui8j2zk4XE96A/E/r5WbuNEueJLlhiV0tCjOBTEpGcRDOlwxakTfEqSK21sJ7aENwtgQXRuCP/nyX3KxW/e9un9m0ziEERZgC7ahBj7swQEcwyk0gcIdPMATPDv3zqPz4ryOWqec8cwm/ILz9g19bJ+i</latexit><latexit sha1_base64="m39V5ZJtlw6UdT+LRypi3+p07tw=">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</latexit><latexit sha1_base64="m39V5ZJtlw6UdT+LRypi3+p07tw=">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</latexit><latexit sha1_base64="WC8J+UKFaLBL3d1QrDN1bhdN9fM=">AAACEXicbVA9SwNBEN2LX/H8ilraLAYhVbiz0UYI2lhYRDAfkAvH3GYvWbJ7d+zuieG4v2DjX7GxUMTWzs5/416SQhMfDDzem2FmXpBwprTjfFulldW19Y3ypr21vbO7V9k/aKs4lYS2SMxj2Q1AUc4i2tJMc9pNJAURcNoJxleF37mnUrE4utOThPYFDCMWMgLaSH6l5g1BCPA9AXokRSbgIbcvbOk72AslkMxLRizPbnK/UnXqzhR4mbhzUkVzNP3KlzeISSpopAkHpXquk+h+BlIzwmlue6miCZAxDGnP0AgEVf1s+lGOT4wywGEsTUUaT9XfExkIpSYiMJ3F4WrRK8T/vF6qw/N+xqIk1TQis0VhyrGOcREPHjBJieYTQ4BIZm7FZAQmCG1CtE0I7uLLy6R9WnedunvrVBuX8zjK6Agdoxpy0RlqoGvURC1E0CN6Rq/ozXqyXqx362PWWrLmM4foD6zPH2mhnVs=</latexit>
⌧max = Gr0�
L<latexit sha1_base64="Ujv2obg7c6rdmbW0GAJ6zH8PH5U=">AAACEXicbVC7SsRAFL3xucZX1NJmUASrJbFxG2HRQguLFdwHbEKYzE7cwZkkzEzEJeQXbPwVGwtFbG3EzspfcfZRqOuBC4dz7uXee6KMM6Vd99OamZ2bX1isLNnLK6tr687GZkuluSS0SVKeyk6EFeUsoU3NNKedTFIsIk7b0fXJ0G/fUKlYmlzqQUYDga8SFjOCtZFCZ9/XOA99gXVfikLg29I+sk+RDF3kxxKTws/6rCzOy9DZdavuCGiaeBOyW699vSMAaITOh99LSS5oognHSnU9N9NBgaVmhNPS9nNFM0yu8RXtGppgQVVQjD4q0Z5ReihOpalEo5H6c6LAQqmBiEzn8HT11xuK/3ndXMe1oGBJlmuakPGiOOdIp2gYD+oxSYnmA0MwkczcikgfmyC0CdE2IXh/X54mrYOq51a9C5PGMYxRgW3YgX3w4BDqcAYNaAKBO3iAJ3i27q1H68V6HbfOWJOZLfgF6+0b+4WfUA==</latexit><latexit sha1_base64="reWKURfheqR1FFMjjBeQYzr3q94=">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</latexit><latexit sha1_base64="reWKURfheqR1FFMjjBeQYzr3q94=">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</latexit><latexit sha1_base64="5eHw0Vm2O4GA2yaWToiLY6NRgcI=">AAACEXicbVC7TsMwFHV4lvAKMLJYVEidqoQFFqQKBhgYikQfUlNFjuu0Vm0nsh1EFeUXWPgVFgYQYmVj429w2gzQcqQrHZ1zr+69J0wYVdp1v62l5ZXVtfXKhr25tb2z6+ztt1WcSkxaOGax7IZIEUYFaWmqGekmkiAeMtIJx5eF37knUtFY3OlJQvocDQWNKEbaSIFT8zVKA58jPZI84+ght8/tKygDF/qRRDjzkxHNs5s8cKpu3Z0CLhKvJFVQohk4X/4gxiknQmOGlOp5bqL7GZKaYkZy208VSRAeoyHpGSoQJ6qfTT/K4bFRBjCKpSmh4VT9PZEhrtSEh6azOF3Ne4X4n9dLdXTWz6hIUk0Eni2KUgZ1DIt44IBKgjWbGIKwpOZWiEfIBKFNiLYJwZt/eZG0T+qeW/du3WrjooyjAg7BEagBD5yCBrgGTdACGDyCZ/AK3qwn68V6tz5mrUtWOXMA/sD6/AHnup0J</latexit>
10
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)
11
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?
12
Torsion