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Qwestrum Engineering360 · Mechanical Engineering · Machine Design

Keys and Couplings

A sunk key is checked for both shear (τ = 2T/(d·l·b)) and crushing (σ_c = 4T/(d·l·h)); the weaker mode fixes the key length. Couplings then transmit that same torque between shafts, with friction couplings using T = μ·F·r·n, following VB Bhandari.
Exam tip: keep units consistent — N with mm² gives MPa directly (1 MPa = 1 N/mm²).

Key formulas & points

Skim these first — then read the full notes below.

  • Saddle key vs sunk key — sunk key more common in practice
  • Design key for lesser of shear or crushing failure
  • Flexible coupling accommodates misalignment; rigid coupling does not

Topic details

Introduction

Keys and couplings appear in Indian design papers as a short-but-scoring question: given shaft torque, size a rectangular sunk key and select the coupling. The key is deliberately made the weakest link so it shears before the shaft or hub fails — a safety-fuse concept examiners like to hear stated.

Scope in B.Tech and GATE syllabus

The candidate must equate the key's shear and crushing capacities to the transmitted torque and take the larger required length. VB Bhandari recommends l ≈ 1.5d as a starting length and cross-checks it against these two failure modes.

Why this topic matters in practice

Couplings — rigid (flange, muff) versus flexible (bush-pin) — are compared on their ability to accommodate misalignment. Flange-coupling bolts are themselves designed in shear and crushing, so the same logic recurs, which is why the topic is examined together.

Key relations & formulas

τkey=2T(d⋅l⋅t)\tau_{key} = \frac{2T}{(d\cdot l\cdot t)}
(shear stress in key, parallel key)
σcrush=2T/(d⋅l⋅(h2))\sigma_{crush} = 2T/(d\cdot l\cdot (\frac{h}{2}))
(crushing stress in key)
T=μ⋅F⋅rT = \mu\cdot F\cdot r
(torque transmitted by friction coupling)
lkey≥1.5dl_{key} \ge 1.5d
(recommended key length, VB Bhandari)

Notation and sign conventions

Relation 1 —
τkey=2T/\tau_{key} = 2T/
τkey=2T(d⋅l⋅t)\tau_{key} = \frac{2T}{(d\cdot l\cdot t)}
(shear stress in key, parallel key)
Write this relation with symbols exactly as in Design of Machine Elements — VB Bhandari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
σcrush=2T/\sigma_{crush} = 2T/
σcrush=2T/(d⋅l⋅(h2))\sigma_{crush} = 2T/(d\cdot l\cdot (\frac{h}{2}))
(crushing stress in key)
Write this relation with symbols exactly as in Design of Machine Elements — VB Bhandari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
T=μ⋅F⋅rT = \mu\cdot F\cdot r
T=μ⋅F⋅rT = \mu\cdot F\cdot r
(torque transmitted by friction coupling)
Write this relation with symbols exactly as in Design of Machine Elements — VB Bhandari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
lkey≥1.5dl_{key} \ge 1.5d
lkey≥1.5dl_{key} \ge 1.5d
(recommended key length, VB Bhandari)
Write this relation with symbols exactly as in Design of Machine Elements — VB Bhandari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Torque on the shaft surface produces a tangential force F = 2T/d at the key. Resisting this over the key's side area (b × l) gives shear stress τ = F/(b·l) = 2T/(d·b·l); resisting it over the crushing area (h/2 × l) gives σ_c = 4T/(d·h·l).

Governing relations in practice

Setting each equal to the allowable value yields two required lengths; the design length is the larger. For a square key (b = h) the crushing stress is twice the shear stress, so crushing usually governs unless the allowable crushing stress is large.

Design and analysis considerations

Rigid couplings assume perfect alignment and simply carry torque; a flange coupling's bolts on a pitch circle of radius R carry F_bolt = 2T/(n·D_pcd) and are sized in shear and crushing just like the key.

Advanced theory and extensions

Friction (cone/disc/clutch) couplings transmit T = μ·F·R_mean·n through the axial clamping force F; here slip torque, not stress, is the limit. Recognising which mechanism transmits the torque is the conceptual key to the whole topic.

Assumptions and validity limits

State assumptions explicitly before using any relation for keys and couplings — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Machine Design viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Machine Design papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to keys and couplings.
4. Use equation 1:
τkey=2T/\tau_{key} = 2T/
.
5. Use equation 2:
σcrush=2T/\sigma_{crush} = 2T/
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Keys and Couplings appears in shafts, keys, bearings, springs, and fasteners. In Indian mechanical curricula this topic is tested because it connects theory to safe sizing of mechanical components.
GATE and semester exams often combine keys and couplings with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use keys and couplings?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Checking only shear of the key and skipping the crushing check (or vice-versa)
• Using full key height h instead of h/2 for the crushing bearing area
• Confusing tangential force F = 2T/d with the torque itself
• Designing flange-coupling bolts in tension when they actually fail in shear/crushing

Quick revision checklist

Before attempting keys and couplings problems, confirm you can:
1. Saddle key vs sunk key — sunk key more common in practice
2. Design key for lesser of shear or crushing failure
3. Flexible coupling accommodates misalignment; rigid coupling does not
Revise the solved examples in Design of Machine Elements — VB Bhandari and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Length of a sunk key

Problem

A 50 mm shaft transmits T = 500 N·m through a rectangular key of width b = 16 mm and height h = 10 mm. Allowable shear stress = 50 MPa. Find the key length for shear failure.

Solution

Tangential force F = 2T/d = 2×500000/50 = 20000 N
l = F/(b·τ) = 20000/(16×50) = 25 mm (shear requirement); also verify crushing before finalising.

Conceptual check — Keys and Couplings

Problem

In a Machine Design semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of keys and couplings." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Keys and Couplings, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    A sunk key is checked for both shear (τ = 2T/(d·l·b)) and crushing (σ_c = 4T/(d·l·h)); the weaker mode fixes the key length. Couplings then transmit that same torque between shafts, with friction couplings using T = μ·F·r·n, following VB Bhandari.
  2. 2
    State the relation τ_key = 2T/ and name each symbol.

    Model answer

    The governing relation is τkey=2T/\tau_{key} = 2T/. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation σ_crush = 2T/ and name each symbol.

    Model answer

    The governing relation is σcrush=2T/\sigma_{crush} = 2T/. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation T = μ·F·r and name each symbol.

    Model answer

    The governing relation is T=μ⋅F⋅rT = \mu\cdot F\cdot r. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation l_key ≥ 1.5d and name each symbol.

    Model answer

    The governing relation is lkey≥1.5dl_{key} \ge 1.5d. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Saddle key vs sunk key — sunk key more common in practice

    Model answer

    Saddle key vs sunk key — sunk key more common in practice — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Design key for lesser of shear or crushing failure

    Model answer

    Design key for lesser of shear or crushing failure — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Flexible coupling accommodates misalignment; rigid coupling does not

    Model answer

    Flexible coupling accommodates misalignment; rigid coupling does not — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Checking only shear of the key and skipping the crushing check (or vice-versa)?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Using full key height h instead of h/2 for the crushing bearing area?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Confusing tangential force F = 2T/d with the torque itself?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Designing flange-coupling bolts in tension when they actually fail in shear/crushing?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    Check both shear and crushing — examiners often set l to fail one criterion.
  • 2
    Avoid: Checking only shear of the key and skipping the crushing check (or vice-versa)
  • 3
    Avoid: Using full key height h instead of h/2 for the crushing bearing area
  • 4
    Avoid: Confusing tangential force F = 2T/d with the torque itself

📖 Standard books (India)

  • Design of Machine Elements — VB Bhandari

    Read: Syllabus unit

    Machine design, shafts, bearings, springs, and joints

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