1 Classes #3 & #4 Civil Engineering Materials – CIVE 2110 Torsion Fall 2010 Dr. Gupta Dr. Pickett

Torque and Torsional Deformation Torque is a MOMENT applied about the LONG axis of a circular shaft. Shaft twists about the LONG axis. Circles remain circles. Each Longitudinal grid line deforms into a HELIX, that intersects the circles at EQUAL angles.

Torque and Torsional Deformation Torque is a MOMENT applied about the LONG axis of a circular shaft. Shaft length & radius do NOT change. Ends of shaft remain flat, perpendicular to long axis. Radial lines remain STRAIGHT.

Torque and Torsional Deformation If shaft is FIXED at one end and Torque applied at other end, Radial lines - remain straight - rotate thru an angle of twist - angle of twist varies along length of shaft

Torque and Torsional Deformation An element located x from the back end of the shaft, will have different angles of twist (rotations) on its front and back faces. The difference in angles of twist causes Torsional SHEAR STRAINS.

Angles change from 90˚.

For all elements on the cross section at x,

For all elements on the cross section at x,

Torsional Shear Stress For circular shafts, hollow or solid, If material is - Homogeneous - Linear-Elastic Hooke’s Law gives: Thus, like Shear Strain, Shear Stress, varies linearly with Radial distance from center of shaft.

Torsional Shear Stress Each elemental area, dA located at ρ from the center, will have an internal Force, dF, that will produce an internal Resisting Torque, T.

Torsional Shear Stress Polar Moment of Inertia, J: for a Solid Shaft;

Axial Shear Stress ( solid shaft ) In order for any element to be in equilibrium the internal torque, T, must develop an Axial Shear Stress, equal to the Radial Shear Stress, also varying Linearly with radial position.

Axial Shear Stress ( hollow shaft ) In order for any element to be in equilibrium the internal torque, T, must develop an Axial Shear Stress, equal to the Radial Shear Stress, also varying Linearly with radial position.

Axial Shear Stress ( wood shaft ) In order for any element to be in equilibrium the internal torque, T, must develop an Axial Shear Stress, equal to the Radial Shear Stress, also varying Linearly with radial position.

Torque and Torsional Deformation If shaft is FIXED at one end and Torque applied at other end, Radial lines - remain straight - rotate thru an angle of twist - angle of twist varies along length of shaft

Torque and Torsional Deformation An element located x from the back end of the shaft, will have different angles of twist (rotations) on its front and back faces. The difference in angles of twist causes Torsional SHEAR STRAINS.

Angle of TWIST Angles change from 90˚.

Angle of TWIST - Angles change from 90˚.

Angle of TWIST

Material must be: - Homogeneous - Isotropic - Linear Elastic - Stress < ( Yield Stress = Proportional Limit )

Angle of TWIST Sign Convention: Positive direction is: - Point right hand thumb outward from shaft - Right hand fingers curl in positive rotation