Chapter11

# Chapter11 - Mechanics& Materials 1 Mechanics&...

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Unformatted text preview: Mechanics & Materials 1 Mechanics & Materials 1 FAMU-FSU College of Engineering Department of Mechanical Engineering Chapter 11 Chapter 11 Torsion Torsion Torsion Torsion • Torsion refers to the twisting of a structural member when it is loaded by couples that produce rotation about the longitudinal axis • The couples that cause the tension are called Torques, Twisting Couples or Twisting Moments Torsion of Circular Bar Torsion of Circular Bar • From consideration of symmetry: – Cross sections of the circular bar rotates as rigid bodies about the longitudinal axis – Cross sections remain straight and circular • If the right hand of the bar rotates through a small angle φ – φ : angle of twist • Line will rotate to a new position • The element ABCD after applying torque d d b b ′ → ′ → Element in state of “pure shear” → Torsion of Circular Bars Torsion of Circular Bars Shear Strain Shear Strain • During torsion, the right-hand cross section of the original configuration of the element (abdc) rotates with respect to the opposite face and points b and c move to b' and c'. • The lengths of the sides of the element do not change during this rotation, but the angles at the corners are no longer 90°. Thus, the element is undergoing pure shear and the magnitude of the shear strain is equal to the decrease in the angle bab'. This angle is ab b b ′ = γ tan Shear Strain Shear Strain γ γ dx rd ab b b φ γ = ′ = rate of change of the angle of twist φ Θ = dx d φ → Angle of twist per unit length → dx d φ tan γ ≈ γ because under pure torsion the angle γ is small. So Under pure torsion, the rate of change d φ /dx of the angle of twist is constant along the length of the bar. This constant is equal to the angle of twist per unit length θ . Θ = = ρ φ ρ γ dx d •Since every cross section is subjected to the same torque so d φ /dx is constant, the shear strain varies along the radial line γ = 0 at ρ = 0 (center of shaft) γ = maximum at ρ = C (outer surface of shaft) ρ γ φ = dx d Shear Strain Shear Strain Shear Strain Shear Strain C dx d max γ φ = const dx = φ d since C dx d max hence γ ρ γ φ = = max γ ρ γ = C The shear strain within the shaft varies linearly along any radial line, from zero at the axis of the shaft to a maximum to at its outer surface Pure Torsion Pure Torsion In the case of pure torsion the rate of change is constant along the length of the bar, because every cross section is subjected to the same torque. Therefore, we obtain dx d φ L r r r Equation L φ θ γ θ γ φ θ = = = = becomes shaft the of length the is L where , THEORY OF TORSION THEORY OF TORSION FORMULA FORMULA • The following conditions are used in the torsion of the circular shaft: 1. Sectional planes perpendicular to the axis of the shaft remain plane during torque application....
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Chapter11 - Mechanics& Materials 1 Mechanics&...

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