Torsion and Shaft Design

Prerequisite knowledge: force equilibrium, SI dimensional analysis, algebra and first-year calculus.

Learning objectives: state a model and its validity range, calculate physically meaningful engineering quantities, reconcile equations with a free-body diagram or deformation compatibility, and identify at least two failure modes excluded by the idealization.

Unit context and engineering interpretation

Torsion arises in drive shafts, fasteners, tools, bridge bearings and structural transfer elements. It is tempting to represent every twisting component by T r/J, but this is rigorously the classical circular-shaft result for linear elastic Saint-Venant torsion. A rectangular section can warp and concentrates stress differently. Thin-walled closed sections transmit shear flow around a cell, while open sections can have very different torsional rigidity. Recognition of the cross-section geometry is therefore the first modeling decision.

Circular-shaft torsional stiffness depends on the polar second moment, which is exceptionally sensitive to diameter. Designers sometimes enlarge a shaft to control rotation rather than peak shear stress. In a stepped shaft, the largest twist may originate in a long thin segment even when another segment carries the peak stress. A coupling or gear introduces local bearing and bending; simply adding a safety factor to nominal T r/J cannot replace a local connection model. The designer should distinguish global torque transmission from the local torque-transfer mechanism.

Power transmission introduces time. Constant mechanical power at a lower angular speed requires higher torque. A rotating shaft may repeatedly experience tension and compression at its outer surface under a stationary transverse load, creating a fatigue scenario absent from static torsional strength. Start-stop cycles, reversal and misalignment complicate the load spectrum. The models in this unit let students control one variable at a time, then identify the omissions in a real coupled drivetrain.

Lesson 4.1 — Circular-shaft torsion in elasticity

Physical mechanism and governing idea

A circular shaft under Saint-Venant torsion develops shear stress that increases linearly with radius in the classical elastic solution. The polar second moment J describes geometric resistance. Torque-angle relationships contain G and J, while stress uses T r/J. These results require a sufficiently circular uniform shaft away from strongly disturbed ends.

Governing relationship

tau(r)=T r/J; theta=T L/(GJ)

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

For a solid circular shaft J=pi d^4/32. Doubling diameter increases J sixteenfold.

What the simplified formula misses

Applying the circular-shaft equation to a thin rectangular bar is invalid; noncircular torsion produces warping and a different stress field.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 4.1 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 4.2 — Hollow shafts and torque transmission

Physical mechanism and governing idea

Removing material near the center of a circular shaft can reduce mass with a smaller penalty in torsional stiffness than removing outer material. For concentric hollow circles, J=pi(do^4-di^4)/32. The maximum shear stress lies at the outer radius. Compare stress, twist, buckling and connection demands, not just mass efficiency.

Governing relationship

J = pi (do^4-di^4)/32; tau_max=T do/(2J)

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

A 60 mm outer diameter and 40 mm inner diameter hollow shaft has a finite J controlled strongly by its outer diameter.

What the simplified formula misses

Thin-walled shafts can buckle locally, and torque transfer at keys or splines creates stress concentrations not captured by uniform shaft theory.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 4.2 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 4.3 — Power, speed and torque

Physical mechanism and governing idea

A motor shaft transports power as angular speed multiplied by torque in consistent SI units. At steady operating speed, T=P/omega. Starting and stopping events can produce torques exceeding the nominal rating. Shaft design must also account for bending from bearings and gears and fatigue from rotating load cycles.

Governing relationship

P=T omega; omega = 2 pi n/60

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

A machine delivering 5 kW at 1500 rpm has a nominal torque of about 31.8 N m.

What the simplified formula misses

Rated power is not a complete load spectrum. Start-up, stall, vibration and impact factors must be established for engineering design.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 4.3 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 4.4 — Angle of twist and compatibility

Physical mechanism and governing idea

A fixed shaft with stepped segments behaves like elastic torsional springs in series when torque is constant. The total twist is the sum T_i L_i/(G_i J_i). A parallel torque path, in contrast, distributes torque according to torsional stiffness when the shafts rotate through the same angle.

Governing relationship

theta_total = sum(T_i L_i/(G_i J_i))

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

Two equal segments in series, one half as stiff as the other, do not share equal twist: the softer segment twists twice as much under the same torque.

What the simplified formula misses

An assembly with non-circular cross sections, splines or compliant couplings may need a more detailed model.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 4.4 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Worked numerical casebook

The following 18 worked cases are intended for dimensional fluency and model interpretation, not as a substitute for unique field measurements. Cases change materials, loads and geometry and deliberately restate their validity conditions.

When reproducing a result, document a complete free body or section cut, record the input units, and calculate at least one quantity without looking at the worked result.

Case 4.01: Nominal torque and twist in a lightweight crane hook

A solid circular shaft transmits 350 N·m torque over 0.8 m, with diameter 35 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=147324 mm⁴. With torque converted to 350000 N·mm, maximum elastic shear is 41.58 MPa at the outer radius. The modeled twist is 1.361 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.01: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.02: Nominal torque and twist in a water-tank frame

A solid circular shaft transmits 600 N·m torque over 1.4 m, with diameter 50 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=613592 mm⁴. With torque converted to 600000 N·mm, maximum elastic shear is 24.45 MPa at the outer radius. The modeled twist is 0.980 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.02: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.03: Nominal torque and twist in a portal-frame member

A solid circular shaft transmits 250 N·m torque over 1.8 m, with diameter 50 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=613592 mm⁴. With torque converted to 250000 N·mm, maximum elastic shear is 10.19 MPa at the outer radius. The modeled twist is 1.616 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.03: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.04: Nominal torque and twist in a cable termination

A solid circular shaft transmits 450 N·m torque over 0.6 m, with diameter 28 mm and shear modulus 40 GPa. Its polar moment is J=πd⁴/32=60344 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 104.40 MPa at the outer radius. The modeled twist is 6.409 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.04: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.05: Nominal torque and twist in a maintenance walkway

A solid circular shaft transmits 450 N·m torque over 1.4 m, with diameter 40 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=251327 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 35.81 MPa at the outer radius. The modeled twist is 1.795 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.05: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.06: Nominal torque and twist in a girder

A solid circular shaft transmits 350 N·m torque over 0.6 m, with diameter 50 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=613592 mm⁴. With torque converted to 350000 N·mm, maximum elastic shear is 14.26 MPa at the outer radius. The modeled twist is 0.245 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.06: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.07: Nominal torque and twist in a roof tie

A solid circular shaft transmits 80 N·m torque over 0.6 m, with diameter 46 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 80000 N·mm, maximum elastic shear is 4.19 MPa at the outer radius. The modeled twist is 0.078 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.07: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.08: Nominal torque and twist in a machine anchor

A solid circular shaft transmits 120 N·m torque over 1.4 m, with diameter 35 mm and shear modulus 75 GPa. Its polar moment is J=πd⁴/32=147324 mm⁴. With torque converted to 120000 N·mm, maximum elastic shear is 14.25 MPa at the outer radius. The modeled twist is 0.871 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.08: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.09: Nominal torque and twist in a laboratory frame

A solid circular shaft transmits 450 N·m torque over 1.4 m, with diameter 22 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=22998 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 215.24 MPa at the outer radius. The modeled twist is 60.367 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.09: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.10: Nominal torque and twist in a small footbridge

A solid circular shaft transmits 250 N·m torque over 1.8 m, with diameter 40 mm and shear modulus 75 GPa. Its polar moment is J=πd⁴/32=251327 mm⁴. With torque converted to 250000 N·mm, maximum elastic shear is 19.89 MPa at the outer radius. The modeled twist is 1.368 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.10: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.11: Nominal torque and twist in a service gantry

A solid circular shaft transmits 80 N·m torque over 0.8 m, with diameter 46 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 80000 N·mm, maximum elastic shear is 4.19 MPa at the outer radius. The modeled twist is 0.321 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.11: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.12: Nominal torque and twist in a construction prop

A solid circular shaft transmits 250 N·m torque over 0.8 m, with diameter 40 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=251327 mm⁴. With torque converted to 250000 N·mm, maximum elastic shear is 19.89 MPa at the outer radius. The modeled twist is 1.754 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.12: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.13: Nominal torque and twist in a testing rig

A solid circular shaft transmits 450 N·m torque over 1.1 m, with diameter 46 mm and shear modulus 75 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 23.55 MPa at the outer radius. The modeled twist is 0.860 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.13: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.14: Nominal torque and twist in a industrial platform

A solid circular shaft transmits 120 N·m torque over 0.6 m, with diameter 28 mm and shear modulus 40 GPa. Its polar moment is J=πd⁴/32=60344 mm⁴. With torque converted to 120000 N·mm, maximum elastic shear is 27.84 MPa at the outer radius. The modeled twist is 1.709 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.14: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.15: Nominal torque and twist in a equipment mount

A solid circular shaft transmits 350 N·m torque over 1.8 m, with diameter 35 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=147324 mm⁴. With torque converted to 350000 N·mm, maximum elastic shear is 41.58 MPa at the outer radius. The modeled twist is 9.424 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.15: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.16: Nominal torque and twist in a maintenance bracket

A solid circular shaft transmits 250 N·m torque over 1.4 m, with diameter 46 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 250000 N·mm, maximum elastic shear is 13.08 MPa at the outer radius. The modeled twist is 1.755 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.16: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.17: Nominal torque and twist in a temporary support

A solid circular shaft transmits 600 N·m torque over 0.8 m, with diameter 46 mm and shear modulus 75 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 600000 N·mm, maximum elastic shear is 31.39 MPa at the outer radius. The modeled twist is 0.834 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.17: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.18: Nominal torque and twist in a pipe rack

A solid circular shaft transmits 450 N·m torque over 1.1 m, with diameter 28 mm and shear modulus 40 GPa. Its polar moment is J=πd⁴/32=60344 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 104.40 MPa at the outer radius. The modeled twist is 11.750 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.18: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Engineering case study: Machine drive shaft with coupling offset

A coupling transmits rated torque between a motor and a gearbox. The maintenance log shows vibration and bearing replacements. Nominal torsion equations give twist and shear for an ideal shaft, but misalignment adds cyclic bending and can shift fatigue hot spots to shoulders or keyways. In a shaft made of multiple diameters, peak torsion stress and maximum total angle of twist need not occur in the same segment.

Deliverable: Draw separate free bodies for torsion and transverse forces. List torque spectrum, shaft dimensions, connection details, bearing positions, alignment measurements and surface quality before discussing a practical retrofit.

Case-study discussion

Start by differentiating data from assumptions: measured loads and geometry should be separated from guessed restraints and material properties. Build one simple calculation whose result can be checked independently. List at least three mechanisms by which the idealization could fail, and prioritize them by whether they would change the engineering conclusion. Translate the numerical demand into a practical question about safety, serviceability, testing or maintenance. Keep a clear audit trail so a reviewer can follow the unit conversions, load path and model limitations.

Calculation verification

• Recompute J directly from diameter rather than rounding a tabulated value too early.
• If torque doubles while diameter and G are fixed, both shear and twist should double in the linear model.
• Check positivity, zero-load behavior, symmetry, and asymptotic scaling whenever applicable.
• Do not use a graph or slider response as the only evidence that an equation is correct.

Structured unit assessment

Answer the following without access to the provided worked solutions. Each question needs a diagram, assumptions and a clear interpretation.

1. Use a load or material value from Case 4.02, change one geometric dimension by 20%, and calculate the resulting response with explicit units.
2. Explain how the result of Lesson 4.1 would change if one apparently rigid support had finite stiffness.
3. Choose a nonuniform or damaged member for which the idealization in Lesson 4.2 becomes inaccurate; describe a better analytical or experimental representation.
4. Derive a useful limiting case from the relationship in Lesson 4.3 and compare with a limiting setting in any associated simulation.
5. A reviewer reports a one-thousandfold discrepancy. Show one plausible N–kN, m–mm or N·m–N·mm conversion error that could create it for a calculation in this unit.
6. For the case study in Unit 4, distinguish two possible failure modes and explain why demonstrating adequate performance against one does not guarantee the other.
7. Create an experimental plan to validate one of the numerical predictions in Unit 4; name the sensor, its calibration requirement and an expected source of uncertainty.
8. State two ways that manufacturing tolerance, support flexibility or temperature could alter a result computed from one equation in Unit 4.

Associated interactive laboratories

Laboratory 03: Circular Shaft Torsion. Within elastic Saint-Venant torsion theory, shear stress increases from zero at the center to the outside surface. Changing diameter alters J dramatically. Examine how diameter affects both peak stress and twist.

Investigation: Double diameter while holding torque and length constant. Estimate the reduction in stress before observing the numerical outputs.

Open the [working HTML model](../labs/lab-03-torsion.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Unit research project

Prepare a technical notebook on Torsion and shaft design for a civil engineering structure of your choice. Include a drawing with geometric dimensions, a carefully numbered input table and a free-body diagram. Develop two load or restraint scenarios, compute the most relevant stress or deformation variables, cross-check them with an independent limiting case, and identify all omissions that would prevent formal design approval. Where an associated laboratory exists, export a CSV series and include an annotated plot. Write a short conclusion that distinguishes what your simplified model genuinely predicts from what would need a professional code-based check.

Project marking rubric

Unit 04 — Torsion and shaft design

Prerequisite knowledge: force equilibrium, SI dimensional analysis, algebra and first-year calculus.

Learning objectives: state a model and its validity range, calculate physically meaningful engineering quantities, reconcile equations with a free-body diagram or deformation compatibility, and identify at least two failure modes excluded by the idealization.

Unit context and engineering interpretation

Torsion arises in drive shafts, fasteners, tools, bridge bearings and structural transfer elements. It is tempting to represent every twisting component by T r/J, but this is rigorously the classical circular-shaft result for linear elastic Saint-Venant torsion. A rectangular section can warp and concentrates stress differently. Thin-walled closed sections transmit shear flow around a cell, while open sections can have very different torsional rigidity. Recognition of the cross-section geometry is therefore the first modeling decision.

Circular-shaft torsional stiffness depends on the polar second moment, which is exceptionally sensitive to diameter. Designers sometimes enlarge a shaft to control rotation rather than peak shear stress. In a stepped shaft, the largest twist may originate in a long thin segment even when another segment carries the peak stress. A coupling or gear introduces local bearing and bending; simply adding a safety factor to nominal T r/J cannot replace a local connection model. The designer should distinguish global torque transmission from the local torque-transfer mechanism.

Power transmission introduces time. Constant mechanical power at a lower angular speed requires higher torque. A rotating shaft may repeatedly experience tension and compression at its outer surface under a stationary transverse load, creating a fatigue scenario absent from static torsional strength. Start-stop cycles, reversal and misalignment complicate the load spectrum. The models in this unit let students control one variable at a time, then identify the omissions in a real coupled drivetrain.

Lesson 4.1 — Circular-shaft torsion in elasticity

Physical mechanism and governing idea

A circular shaft under Saint-Venant torsion develops shear stress that increases linearly with radius in the classical elastic solution. The polar second moment J describes geometric resistance. Torque-angle relationships contain G and J, while stress uses T r/J. These results require a sufficiently circular uniform shaft away from strongly disturbed ends.

Governing relationship

tau(r)=T r/J; theta=T L/(GJ)

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

For a solid circular shaft J=pi d^4/32. Doubling diameter increases J sixteenfold.

What the simplified formula misses

Applying the circular-shaft equation to a thin rectangular bar is invalid; noncircular torsion produces warping and a different stress field.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 4.1 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 4.2 — Hollow shafts and torque transmission

Physical mechanism and governing idea

Removing material near the center of a circular shaft can reduce mass with a smaller penalty in torsional stiffness than removing outer material. For concentric hollow circles, J=pi(do^4-di^4)/32. The maximum shear stress lies at the outer radius. Compare stress, twist, buckling and connection demands, not just mass efficiency.

Governing relationship

J = pi (do^4-di^4)/32; tau_max=T do/(2J)

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

A 60 mm outer diameter and 40 mm inner diameter hollow shaft has a finite J controlled strongly by its outer diameter.

What the simplified formula misses

Thin-walled shafts can buckle locally, and torque transfer at keys or splines creates stress concentrations not captured by uniform shaft theory.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 4.2 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 4.3 — Power, speed and torque

Physical mechanism and governing idea

A motor shaft transports power as angular speed multiplied by torque in consistent SI units. At steady operating speed, T=P/omega. Starting and stopping events can produce torques exceeding the nominal rating. Shaft design must also account for bending from bearings and gears and fatigue from rotating load cycles.

Governing relationship

P=T omega; omega = 2 pi n/60

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

A machine delivering 5 kW at 1500 rpm has a nominal torque of about 31.8 N m.

What the simplified formula misses

Rated power is not a complete load spectrum. Start-up, stall, vibration and impact factors must be established for engineering design.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 4.3 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 4.4 — Angle of twist and compatibility

Physical mechanism and governing idea

A fixed shaft with stepped segments behaves like elastic torsional springs in series when torque is constant. The total twist is the sum T_i L_i/(G_i J_i). A parallel torque path, in contrast, distributes torque according to torsional stiffness when the shafts rotate through the same angle.

Governing relationship

theta_total = sum(T_i L_i/(G_i J_i))

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

Two equal segments in series, one half as stiff as the other, do not share equal twist: the softer segment twists twice as much under the same torque.

What the simplified formula misses

An assembly with non-circular cross sections, splines or compliant couplings may need a more detailed model.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 4.4 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Worked numerical casebook

The following 18 worked cases are intended for dimensional fluency and model interpretation, not as a substitute for unique field measurements. Cases change materials, loads and geometry and deliberately restate their validity conditions.

When reproducing a result, document a complete free body or section cut, record the input units, and calculate at least one quantity without looking at the worked result.

Case 4.01: Nominal torque and twist in a lightweight crane hook

A solid circular shaft transmits 350 N·m torque over 0.8 m, with diameter 35 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=147324 mm⁴. With torque converted to 350000 N·mm, maximum elastic shear is 41.58 MPa at the outer radius. The modeled twist is 1.361 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.01: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.02: Nominal torque and twist in a water-tank frame

A solid circular shaft transmits 600 N·m torque over 1.4 m, with diameter 50 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=613592 mm⁴. With torque converted to 600000 N·mm, maximum elastic shear is 24.45 MPa at the outer radius. The modeled twist is 0.980 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.02: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.03: Nominal torque and twist in a portal-frame member

A solid circular shaft transmits 250 N·m torque over 1.8 m, with diameter 50 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=613592 mm⁴. With torque converted to 250000 N·mm, maximum elastic shear is 10.19 MPa at the outer radius. The modeled twist is 1.616 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.03: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.04: Nominal torque and twist in a cable termination

A solid circular shaft transmits 450 N·m torque over 0.6 m, with diameter 28 mm and shear modulus 40 GPa. Its polar moment is J=πd⁴/32=60344 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 104.40 MPa at the outer radius. The modeled twist is 6.409 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.04: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.05: Nominal torque and twist in a maintenance walkway

A solid circular shaft transmits 450 N·m torque over 1.4 m, with diameter 40 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=251327 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 35.81 MPa at the outer radius. The modeled twist is 1.795 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.05: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.06: Nominal torque and twist in a girder

A solid circular shaft transmits 350 N·m torque over 0.6 m, with diameter 50 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=613592 mm⁴. With torque converted to 350000 N·mm, maximum elastic shear is 14.26 MPa at the outer radius. The modeled twist is 0.245 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.06: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.07: Nominal torque and twist in a roof tie

A solid circular shaft transmits 80 N·m torque over 0.6 m, with diameter 46 mm and shear modulus 80 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 80000 N·mm, maximum elastic shear is 4.19 MPa at the outer radius. The modeled twist is 0.078 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.07: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.08: Nominal torque and twist in a machine anchor

A solid circular shaft transmits 120 N·m torque over 1.4 m, with diameter 35 mm and shear modulus 75 GPa. Its polar moment is J=πd⁴/32=147324 mm⁴. With torque converted to 120000 N·mm, maximum elastic shear is 14.25 MPa at the outer radius. The modeled twist is 0.871 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.08: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.09: Nominal torque and twist in a laboratory frame

A solid circular shaft transmits 450 N·m torque over 1.4 m, with diameter 22 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=22998 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 215.24 MPa at the outer radius. The modeled twist is 60.367 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.09: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.10: Nominal torque and twist in a small footbridge

A solid circular shaft transmits 250 N·m torque over 1.8 m, with diameter 40 mm and shear modulus 75 GPa. Its polar moment is J=πd⁴/32=251327 mm⁴. With torque converted to 250000 N·mm, maximum elastic shear is 19.89 MPa at the outer radius. The modeled twist is 1.368 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.10: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.11: Nominal torque and twist in a service gantry

A solid circular shaft transmits 80 N·m torque over 0.8 m, with diameter 46 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 80000 N·mm, maximum elastic shear is 4.19 MPa at the outer radius. The modeled twist is 0.321 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.11: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.12: Nominal torque and twist in a construction prop

A solid circular shaft transmits 250 N·m torque over 0.8 m, with diameter 40 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=251327 mm⁴. With torque converted to 250000 N·mm, maximum elastic shear is 19.89 MPa at the outer radius. The modeled twist is 1.754 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.12: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.13: Nominal torque and twist in a testing rig

A solid circular shaft transmits 450 N·m torque over 1.1 m, with diameter 46 mm and shear modulus 75 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 23.55 MPa at the outer radius. The modeled twist is 0.860 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.13: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.14: Nominal torque and twist in a industrial platform

A solid circular shaft transmits 120 N·m torque over 0.6 m, with diameter 28 mm and shear modulus 40 GPa. Its polar moment is J=πd⁴/32=60344 mm⁴. With torque converted to 120000 N·mm, maximum elastic shear is 27.84 MPa at the outer radius. The modeled twist is 1.709 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.14: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.15: Nominal torque and twist in a equipment mount

A solid circular shaft transmits 350 N·m torque over 1.8 m, with diameter 35 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=147324 mm⁴. With torque converted to 350000 N·mm, maximum elastic shear is 41.58 MPa at the outer radius. The modeled twist is 9.424 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.15: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.16: Nominal torque and twist in a maintenance bracket

A solid circular shaft transmits 250 N·m torque over 1.4 m, with diameter 46 mm and shear modulus 26 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 250000 N·mm, maximum elastic shear is 13.08 MPa at the outer radius. The modeled twist is 1.755 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.16: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.17: Nominal torque and twist in a temporary support

A solid circular shaft transmits 600 N·m torque over 0.8 m, with diameter 46 mm and shear modulus 75 GPa. Its polar moment is J=πd⁴/32=439573 mm⁴. With torque converted to 600000 N·mm, maximum elastic shear is 31.39 MPa at the outer radius. The modeled twist is 0.834 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.17: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 4.18: Nominal torque and twist in a pipe rack

A solid circular shaft transmits 450 N·m torque over 1.1 m, with diameter 28 mm and shear modulus 40 GPa. Its polar moment is J=πd⁴/32=60344 mm⁴. With torque converted to 450000 N·mm, maximum elastic shear is 104.40 MPa at the outer radius. The modeled twist is 11.750 degrees. Coupling effects and cyclic bending are outside this uniform torsion calculation.

Extension task 4.18: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Engineering case study: Machine drive shaft with coupling offset

A coupling transmits rated torque between a motor and a gearbox. The maintenance log shows vibration and bearing replacements. Nominal torsion equations give twist and shear for an ideal shaft, but misalignment adds cyclic bending and can shift fatigue hot spots to shoulders or keyways. In a shaft made of multiple diameters, peak torsion stress and maximum total angle of twist need not occur in the same segment.

Deliverable: Draw separate free bodies for torsion and transverse forces. List torque spectrum, shaft dimensions, connection details, bearing positions, alignment measurements and surface quality before discussing a practical retrofit.

Case-study discussion

Start by differentiating data from assumptions: measured loads and geometry should be separated from guessed restraints and material properties. Build one simple calculation whose result can be checked independently. List at least three mechanisms by which the idealization could fail, and prioritize them by whether they would change the engineering conclusion. Translate the numerical demand into a practical question about safety, serviceability, testing or maintenance. Keep a clear audit trail so a reviewer can follow the unit conversions, load path and model limitations.

Calculation verification

• Recompute J directly from diameter rather than rounding a tabulated value too early.
• If torque doubles while diameter and G are fixed, both shear and twist should double in the linear model.
• Check positivity, zero-load behavior, symmetry, and asymptotic scaling whenever applicable.
• Do not use a graph or slider response as the only evidence that an equation is correct.

Structured unit assessment

Answer the following without access to the provided worked solutions. Each question needs a diagram, assumptions and a clear interpretation.

1. Use a load or material value from Case 4.02, change one geometric dimension by 20%, and calculate the resulting response with explicit units.
2. Explain how the result of Lesson 4.1 would change if one apparently rigid support had finite stiffness.
3. Choose a nonuniform or damaged member for which the idealization in Lesson 4.2 becomes inaccurate; describe a better analytical or experimental representation.
4. Derive a useful limiting case from the relationship in Lesson 4.3 and compare with a limiting setting in any associated simulation.
5. A reviewer reports a one-thousandfold discrepancy. Show one plausible N–kN, m–mm or N·m–N·mm conversion error that could create it for a calculation in this unit.
6. For the case study in Unit 4, distinguish two possible failure modes and explain why demonstrating adequate performance against one does not guarantee the other.
7. Create an experimental plan to validate one of the numerical predictions in Unit 4; name the sensor, its calibration requirement and an expected source of uncertainty.
8. State two ways that manufacturing tolerance, support flexibility or temperature could alter a result computed from one equation in Unit 4.

Associated interactive laboratories

Laboratory 03: Circular Shaft Torsion. Within elastic Saint-Venant torsion theory, shear stress increases from zero at the center to the outside surface. Changing diameter alters J dramatically. Examine how diameter affects both peak stress and twist.

Investigation: Double diameter while holding torque and length constant. Estimate the reduction in stress before observing the numerical outputs.

Open the [working HTML model](../labs/lab-03-torsion.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Unit research project

Prepare a technical notebook on Torsion and shaft design for a civil engineering structure of your choice. Include a drawing with geometric dimensions, a carefully numbered input table and a free-body diagram. Develop two load or restraint scenarios, compute the most relevant stress or deformation variables, cross-check them with an independent limiting case, and identify all omissions that would prevent formal design approval. Where an associated laboratory exists, export a CSV series and include an annotated plot. Write a short conclusion that distinguishes what your simplified model genuinely predicts from what would need a professional code-based check.

Project marking rubric

---