🔗Statics and Strength of Materials Unit 10 – Torsion

Key Concepts and Definitions

• Torsion occurs when a member is subjected to a twisting load that produces shear stresses and strains
• Shear stress $(\tau)$ is the force per unit area acting parallel to the surface of a material
• Shear strain $(\gamma)$ is the angular deformation caused by shear stress
• Torque $(T)$ is the twisting moment that causes torsional deformation
• Polar moment of inertia $(J)$ is a measure of a cross-section's resistance to torsion
  • Depends on the shape and dimensions of the cross-section
• Modulus of rigidity $(G)$ is a material property that relates shear stress to shear strain
• Torsional stiffness $(GJ)$ is a measure of a member's resistance to torsional deformation
• Angle of twist $(\phi)$ is the angular rotation of a cross-section due to torsional loading

Torsion Theory and Principles

• Torsion theory is based on the assumption that cross-sections remain plane and undistorted during torsional deformation
• Shear stresses in torsion vary linearly from zero at the center of the cross-section to a maximum at the outer surface
• The maximum shear stress occurs at the point farthest from the centroid of the cross-section
• Torsional deformation is proportional to the applied torque and inversely proportional to the torsional stiffness
• The angle of twist is constant along the length of a member with a uniform cross-section and constant torque
• Torsional shear stresses act in the plane of the cross-section and are perpendicular to the radius
• The distribution of shear stresses in a cross-section depends on its shape and dimensions
  • Circular cross-sections have a uniform shear stress distribution

Types of Torsional Loading

• Pure torsion occurs when a member is subjected to torques that act about its longitudinal axis
• Combined torsion and bending occur when a member is subjected to both torsional and bending loads simultaneously
• Statically determinate torsional loading occurs when the torques acting on a member can be determined from static equilibrium equations alone
• Statically indeterminate torsional loading occurs when additional compatibility equations are required to determine the torques acting on a member
• Distributed torsional loading occurs when torques are applied along the length of a member (non-uniform torsion)
• Concentrated torsional loading occurs when torques are applied at discrete points along the length of a member
• Torsional loading can be applied in a clockwise or counterclockwise direction
  • The sign convention for torques depends on the chosen coordinate system

Stress and Strain in Torsion

• Shear stress in torsion is proportional to the distance from the centroid of the cross-section
  • $\tau = \frac{Tr}{J}$, where $r$ is the distance from the centroid
• The maximum shear stress occurs at the outer surface of the cross-section
  • $\tau_{max} = \frac{Tc}{J}$, where $c$ is the distance from the centroid to the outer surface
• Shear strain in torsion is related to the angle of twist per unit length
  • $\gamma = r\frac{d\phi}{dx}$, where $\frac{d\phi}{dx}$ is the rate of change of the angle of twist along the member
• The relationship between shear stress and shear strain is given by Hooke's law for shear
  • $\tau = G\gamma$, where $G$ is the modulus of rigidity
• Torsional stress and strain are proportional to the applied torque and inversely proportional to the polar moment of inertia
• The distribution of shear stresses and strains in a cross-section depends on its shape and dimensions
• Torsional stress and strain can cause yielding, plastic deformation, or failure of a material if they exceed the material's strength limits

Torsional Deformation and Angle of Twist

• Torsional deformation is the twisting of a member about its longitudinal axis due to applied torques
• The angle of twist is the angular rotation of a cross-section relative to its original position
• The angle of twist is proportional to the applied torque, length of the member, and inversely proportional to the torsional stiffness
  • $\phi = \frac{TL}{GJ}$, where $L$ is the length of the member
• The rate of change of the angle of twist along the member is constant for uniform torsion
  • $\frac{d\phi}{dx} = \frac{T}{GJ}$
• Torsional deformation can cause warping of non-circular cross-sections
  • Warping is the out-of-plane distortion of a cross-section due to torsion
• Restrained warping occurs when the warping of a cross-section is prevented, leading to additional stresses and deformations
• Torsional deformation can be reduced by increasing the torsional stiffness of a member
  • This can be achieved by using materials with higher moduli of rigidity or increasing the polar moment of inertia of the cross-section

Material Properties and Behavior

• The modulus of rigidity $(G)$ is a material property that relates shear stress to shear strain
  • It is a measure of a material's resistance to shear deformation
• The modulus of rigidity is related to the modulus of elasticity $(E)$ and Poisson's ratio $(\nu)$
  • $G = \frac{E}{2(1 + \nu)}$
• Typical values of the modulus of rigidity for common engineering materials:
  • Steel: 75-80 GPa
  • Aluminum: 25-30 GPa
  • Copper: 40-50 GPa
• The yield strength in shear $(\tau_y)$ is the shear stress at which a material begins to exhibit plastic deformation
  • It is typically 50-60% of the yield strength in tension $(\sigma_y)$
• The ultimate strength in shear $(\tau_u)$ is the maximum shear stress a material can withstand before failure
• Ductile materials (steel, aluminum) can undergo significant plastic deformation before failure in torsion
• Brittle materials (cast iron, concrete) have limited capacity for plastic deformation and can fail suddenly in torsion
• Fatigue failure can occur in materials subjected to cyclic torsional loading
  • Fatigue strength is the maximum stress a material can withstand for a given number of loading cycles

Torsion Equations and Calculations

• Torsional stress: $\tau = \frac{Tr}{J}$
  • Maximum torsional stress: $\tau_{max} = \frac{Tc}{J}$
• Angle of twist: $\phi = \frac{TL}{GJ}$
• Rate of change of angle of twist: $\frac{d\phi}{dx} = \frac{T}{GJ}$
• Torsional stiffness: $k_t = \frac{GJ}{L}$
• Polar moment of inertia for solid circular shafts: $J = \frac{\pi d^4}{32}$
  • $d$ is the diameter of the shaft
• Polar moment of inertia for thin-walled tubes: $J = \frac{\pi}{32}(D^4 - d^4)$
  • $D$ is the outer diameter, and $d$ is the inner diameter
• Torsional strain energy: $U = \frac{T^2L}{2GJ}$
• Power transmitted by a rotating shaft: $P = T\omega$
  • $\omega$ is the angular velocity in radians per second
• Equivalent twisting moment for combined torsion and bending: $T_e = \sqrt{T^2 + (\alpha M)^2}$
  • $M$ is the bending moment, and $\alpha$ is a factor depending on the material and cross-section

Applications and Real-World Examples

• Torsion is a common loading condition in power transmission shafts (vehicle driveshafts, industrial machinery)
• Torsional vibration can occur in rotating machinery, leading to fatigue failure and noise
  • Torsional vibration dampers are used to reduce vibrations and extend the life of components
• Torsion springs store energy through torsional deformation and are used in various applications (door hinges, garage doors, watches)
• Torsional stiffness is an important consideration in the design of automobile chassis and suspension components
  • Higher torsional stiffness improves handling and reduces body roll during cornering
• Torsion is a key factor in the design of bolted joints and threaded fasteners
  • Tightening torques must be carefully controlled to prevent overloading and failure
• Torsional loading can cause failure in structural members (bridge girders, wind turbine blades) if not properly accounted for in the design
• Torsion testing is used to determine the shear properties of materials and to evaluate the performance of components subjected to torsional loading
• Finite element analysis (FEA) is commonly used to analyze torsional stresses and deformations in complex geometries and loading conditions