CELE Strength of Materials — TorsionConcept Map

Concept

Torsion in Circular Shafts

Sub Concepts

• Shear stress distribution (linear from center to surface)
• Maximum shear stress formula: τ_max = Tc/J
• Solid shaft formula: τ_max = 16T/(πd³)
• Polar moment of inertia (J) for circular sections
• Hollow shaft polar moment: J = π(D⁴-d⁴)/32

Relationship To Central

Core theory and primary application

Concept

Angle of Twist

Sub Concepts

• Relative rotation between two sections
• Formula: θ = TL/(JG) in radians
• Effect of torque magnitude on angle
• Effect of shaft length on twist
• Effect of shear modulus G on rigidity
• Stepped shafts: sum of individual segments
• Conversion to degrees for reporting

Relationship To Central

Deformation response to torsional loading

Concept

Power Transmission

Sub Concepts

• Relationship: P = 2πfT (f in rev/s)
• Rotational speed in rpm: P = 2πNT/60
• Torque calculation from power and speed
• Two-step problem-solving: power → torque → stress
• Units: watts, N·m, rpm, rev/s

Relationship To Central

Practical engineering application linking power to torque

Concept

Flanged Bolt Couplings

Sub Concepts

• Single bolt circle: T = Pℝn = Aτℝn
• Shear force per bolt: P = Aτ
• Two concentric circles: force proportional to radius
• Bolt compatibility condition: P₁/R₁ = P₂/R₂
• Total torque: T = (P₁R₁n₁) + (P₂R₂n₂)

Relationship To Central

Design of practical torque transmission devices

Concept

Thin-Walled Closed Tubes

Sub Concepts

• Shear flow concept: q = τt (constant around perimeter)
• Stress formula: τ = T/(2A_m·t)
• Median (centerline) area A_m definition
• Constant shear flow around section
• Application to hollow boxes and tubes

Relationship To Central

Extension to non-circular hollow sections

Concept

Material Properties & Assumptions

Sub Concepts

• Linear elasticity: stress proportional to strain
• Homogeneous material
• Circular cross-section remains plane (no warping)
• Stress within proportional limit
• Shear modulus G for steel ≈ 80 GPa

Relationship To Central

Foundation for torsion theory validity

Concept

Shear Stress & Strain Distribution

Sub Concepts

• Linear stress variation: τ = Tρ/J
• Zero stress at center, maximum at surface
• Shear strain: γ = τ/G = cθ/L
• Strain compatibility with deformation
• Mohr's circle for combined loading

Relationship To Central

Physical behavior of material under torsion

Concept

Combined Loading

Sub Concepts

• Bending moment M + Torque T combination
• Axial load + Torsion combination
• Principal stresses via Mohr's circle
• Equivalent torque: T_e = √(M² + T²)
• Equivalent moment: M_e = ½(M + √(M²+T²))

Relationship To Central

Real-world scenario of torque plus other forces

Concept

Design & Sizing

Sub Concepts

• Allowable shear stress selection
• Diameter calculation from stress limit
• Torque capacity determination
• Factor of safety considerations
• Standard size selection (practical application)

Relationship To Central

Engineering application for shaft selection

To

Polar Moment of Inertia (J)

From

Torsion Formula (τ = Tc/J)

Strength

strong

Relationship

Torsion formula depends directly on J; J is the denominator and critical to stress calculation

To

Maximum Shear Stress

From

Torsion Formula (τ = Tc/J)

Strength

strong

Relationship

Formula directly calculates maximum shear stress at outer radius c

To

Torsion Formula

From

Power Transmission

Strength

strong

Relationship

Power and speed determine torque; torque is input to torsion formula for stress

To

Shear Modulus G

From

Angle of Twist Formula

Strength

strong

Relationship

G appears in denominator of θ = TL/(JG); higher G means less twist

To

Torsion Formula

From

Angle of Twist Formula

Strength

moderate

Relationship

Both use polar moment J; both dependent on material and geometry

To

Torsion Formula

From

Flanged Bolt Couplings

Strength

moderate

Relationship

Coupling bolts carry shear stress; maximum stress in bolts must satisfy torsion limits

To

Torsion Formula

From

Thin-Walled Tubes

Strength

moderate

Relationship

Alternative approach using shear flow; applies when wall thickness is small

To

Torsion Formula

From

Combined Loading (M + T)

Strength

moderate

Relationship

Torque stress component calculated via torsion formula; combined with bending stress

To

Linear Elasticity Assumption

From

Shear Stress Distribution

Strength

strong

Relationship

Linear stress distribution requires Hooke's Law and stress within proportional limit

To

Power Transmission

From

Shaft Sizing (Design)

Strength

strong

Relationship

Power and speed input; diameter output to carry power safely

To

Hollow Shaft Efficiency

From

Solid Shaft Formula τ = 16T/(πd³)

Strength

moderate

Relationship

Solid formula shows why hollow shafts (same outer d, removed core) reduce weight for same torque

To

Shear Strain

From

Angle of Twist

Strength

moderate

Relationship

Twist angle θ produces surface shear strain γ = cθ/L; related via geometry

To

Design and Sizing

From

Allowable Stress

Strength

strong

Relationship

Allowable shear stress τ_allow is limit used to size shaft diameter

To

Circular Shaft Theory

From

Standard Cross-Section Assumption

Strength

strong

Relationship

Theory assumes circular section remains plane and circular; limits applicability