CIVE 1160 Ch05

CIVE 1160 Ch05 - College of Engineering CIVE 1160 Summer...

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Unformatted text preview: College of Engineering CIVE 1160 Summer 2011 Chapter 5 Torsion Big ideas n Based on assumptions that are validated by experiment, we will develop w t = f( load, geometry) w q = f( load, geometry, material) College of Engineering CIVE 1160 Summer 2011 Torsion Shafts are long, circular members that are loaded in torsion. Torque is a twisting moment acting about the longitudinal axis of shafts. Torsion is a term that is used to describe the loading that develops College of Engineering CIVE 1160 Summer 2011 5.1 Torsional Deformation Described To derive an expression for the strain, it is necessary to observe what deformations occur. The following observations are given for shafts loaded in torsion. Cross-sectional areas remain circular. Longitudinal lines deform into College of Engineering CIVE 1160 Summer 2011 5.1 Torsional Deformation Fig. 5-1 College of Engineering CIVE 1160 Summer 2011 Shear Stress and Strain Before After The torque or twisting moment acts parallel to sections taken perpendicular to the longitudinal axis. Therefore, shear stress develops in on the sections. The shear stress results in shear strain . The deformation associated with shear strain is College of Engineering CIVE 1160 Summer 2011 Angle of Twist The angle of twist, φ (x), is the amount of rotation that occurs in a shaft. Fig. 5-2 Fixed End College of Engineering CIVE 1160 Summer 2011 Shear Strain Shear Strain is defined as the change in angle between two segments that are originally perpendicular. Shear strain is determined by finding the change in angle that occurs over length dx. θ π γ ′ → →- = l i m 2 A B A C College of Engineering CIVE 1160 Summer 2011 Difference in Rotation For an element of length ∆ x, the rotation at distance x is φ (x) and the rotation at distance x + ∆ x is φ (x)+ ∆φ . Fig. 5-3 College of Engineering CIVE 1160 Summer 2011 Shear Strain at Radial Distance ρ Fig. 5-3 d x d φ ρ γ = ⊥ Before loading, AB AC. Upon loading, pt. B rotates to pt. D. Letting ∆ x dx and ∆φ d φ : BD = ρ d φ and BD = dx tan γ = dx College of Engineering CIVE 1160 Summer 2011 Radial Variation of Shear Strain From the previous equation: Shear strain varies linearly from zero at the axis of the shaft to a maximum at the outside radius, c. m a x m a x γ ρ γ γ ρ γ φ = = = c c d x d College of Engineering CIVE 1160 Summer 2011 5.1 Radial Variation of Shear Stress For linear, elastic behavior: and Multiplying the previous equation by G: Shear stress varies linearly from zero at the axis of the shaft to a maximum at the outside radius, c. γ τ ∆ ∆ = G γ τ G = m ax m ax G τ ρ τ γ ρ γ = = c c G College of Engineering CIVE 1160 Summer 2011 5.2 Relating Torque and Stress • We have an equation for the distribution of shear stress throughout the shaft, but what we want is a relationship between the applied...
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CIVE 1160 Ch05 - College of Engineering CIVE 1160 Summer...

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