Torsion - Concept Map
Torsion: Twisting of Members Under Torque
Related concepts
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
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
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
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
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
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
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
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
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
Concept connections
- Torsion Formula (τ = Tc/J)Polar Moment of Inertia (J)Torsion formula depends directly on J; J is the denominator and critical to stress calculation
- Torsion Formula (τ = Tc/J)Maximum Shear StressFormula directly calculates maximum shear stress at outer radius c
- Power TransmissionTorsion FormulaPower and speed determine torque; torque is input to torsion formula for stress
- Angle of Twist FormulaShear Modulus GG appears in denominator of θ = TL/(JG); higher G means less twist
- Angle of Twist FormulaTorsion FormulaBoth use polar moment J; both dependent on material and geometry
- Flanged Bolt CouplingsTorsion FormulaCoupling bolts carry shear stress; maximum stress in bolts must satisfy torsion limits
- Thin-Walled TubesTorsion FormulaAlternative approach using shear flow; applies when wall thickness is small
- Combined Loading (M + T)Torsion FormulaTorque stress component calculated via torsion formula; combined with bending stress
- Shear Stress DistributionLinear Elasticity AssumptionLinear stress distribution requires Hooke's Law and stress within proportional limit
- Shaft Sizing (Design)Power TransmissionPower and speed input; diameter output to carry power safely
- Solid Shaft Formula τ = 16T/(πd³)Hollow Shaft EfficiencySolid formula shows why hollow shafts (same outer d, removed core) reduce weight for same torque
- Angle of TwistShear StrainTwist angle θ produces surface shear strain γ = cθ/L; related via geometry
- Allowable StressDesign and SizingAllowable shear stress τ_allow is limit used to size shaft diameter
- Standard Cross-Section AssumptionCircular Shaft TheoryTheory assumes circular section remains plane and circular; limits applicability
Diagrams
Torsion - Complete Concept Hierarchy
Comprehensive mind map showing all key concepts in Torsion, organized hierarchically from the central concept of torsion through circular shaft theory, deformation, power transmission, practical devices, and design applications. Shows how polar moment of inertia, shear stress distribution, angle of twist, and power transmission relate to one another.
Torsion Problem-Solving Flowchart
Decision tree and step-by-step flowchart for solving the four main types of torsion problems: (1) maximum shear stress in shafts, (2) angle of twist, (3) shaft diameter from power transmission, and (4) torque capacity of flanged couplings. Guides the solver through data collection, formula selection, unit conversion, and result reporting.
Polar Moment of Inertia (J) Calculation Decision Tree
Decision tree for selecting and calculating the polar moment of inertia. Distinguishes between solid circular shafts, hollow circular shafts, and thin-walled tubes. Highlights the critical error of subtracting areas instead of fourth powers for hollow shafts, and notes that thin-walled tubes use median area instead of J.
Power Transmission to Shaft Diameter - Standard Board Workflow
Step-by-step workflow for the most common board exam problem: given power, speed, and allowable stress, find the minimum solid shaft diameter. This is a two-step calculation (power to torque, then torque to diameter) with critical unit management and standard size selection at the end.
Torsion with Combined Loading - Stress Analysis
Complete workflow for analyzing shafts under combined bending and torsional loading. Shows how to compute individual stresses, find principal stresses using Mohr's circle, apply failure criteria (Tresca or Von Mises), and perform design verification using equivalent torque.
Torsion Load States and Material Response
State diagram showing the material response to torsional loading, from unloaded through elastic deformation (where torsion formulas apply) through the proportional limit into plastic deformation (where formulas fail) and ultimately to failure. Illustrates why staying within the elastic range is critical for formula validity.
Typical Distribution of Torsion Problems on PRC Board Exams
Distribution of typical torsion problem types encountered in PRC Civil Engineer Licensure Examination. Reflects that maximum shear stress and power transmission dominate the exams (65% combined), with angle of twist and couplings as secondary topics. This helps prioritize study effort.