Mechanical Design Data and Machine-Element Reference: Maximum Bearing Pressure Reference

Maximum Bearing Pressure Reference

Worked Example: Porous Bronze Bearing Selection

• Given: 30 mm diameter shaft, 1450 rev/min, 500 N total radial load shared equally between two bearings → 250 N per bearing, continuous running
• Step 1: Load per bearing = 500/2 = 250 N
• Step 2: Choose 30 mm bearing length (L/d = 1)
• Step 3: p = 250 / (30 × 30) = 0.278 MPa
• Step 4: v = (30/2000) × (2π × 1450/60) = 2.278 m/s
• Step 5: Since v > 1 m/s, use the pressure vs. speed chart → for 30 mm at 1450 rev/min, maximum pressure = 0.47 MPa → 0.278 < 0.47 → OK
• Step 6: p·v = 0.278 × 2.278 = 0.633 → exceeds 0.53 → auxiliary lubrication required
• Step 7: M = (20 × 2.278) / 0.278 = 164 → well above 75 → thick-film lubrication assured
• Step 8: Selected bearing has a nominal inside diameter of 30 mm and outside diameter of 38 mm with a length of 30 mm; auxiliary lubrication (e.g., reservoir with felt washer) is required

Auxiliary Lubrication Methods

• Felt washer soaked in oil with a steel retainer
• Felt wick and oil well arrangement
• Oil reservoir and felt washer (or wool) surrounding the bearing
• Felt pad with spring pressure and a screw cap filled with light grease

Standard Metric Bearing Size Tables

Cylindrical Bearings (Standard Sizes)

Flange Bearings (Standard Sizes)

• Flange bearings have a flange with a specified flange diameter and flange thickness to allow thrust load support and ease of mounting
• Available from 12 mm to 70 mm flange diameter with thicknesses from 2 mm to 5 mm

Non-Standard Metric Cylindrical Bearings

• Available for nominal inside diameters from 3 mm to 25 mm
• Outside diameters range from 5 mm to 30 mm
• Lengths range from 5 mm to 50 mm

Belt Drives

Types of Belts

• Vee belts: older type of V-shaped design; now largely superseded by wedge belts but the term "vee belt" is still commonly used for both types
• Wedge belts: similar to vee belts but with a deeper profile; slightly more expensive but have higher power ratings; now more commonly specified for new designs; pulleys are interchangeable with vee belt pulleys
• Banded belts: multiple belts joined together at the top to form a composite belt; eliminates belt twist, whip, and turnover
• Multi-pull belts: similar to banded belts but with a different profile
• Link belts: hybrid between vee belts and chain drives; made of reinforced urethane linked sections; resistant to heat, oil, and chemicals; can be made to any length; uses the same pulleys as vee/wedge belts
• Cogged raw edge (CRE) belts: similar to vee belts but with a cogged edge on the underside; can be used with smaller pulleys; commonly used in automotive and high-speed drives; same pulleys as vee/wedge belts
• Synchronous belts: flat belts with a toothed profile that mates with gear teeth on the pulleys; prevents belt slip and maintains pulley synchronisation; lower load capacity than vee or wedge belts

Key Design Notes

• Modern design data is given for wedge belts (not vee belts)
• Pulleys use a taper lock design — when specifying, both the pulley catalogue number and the bush catalogue number are needed
• Belt designation tables give belt length, combined arc of contact, and belt length correction factor
• No specific national standard exists for vee or wedge belts, but commercial products comply with both relevant national and international standards

Wedge Belt Selection and Drive Design Procedure

1. Determine the service factor from the service factor table based on:

   • Type of driven machine (light duty through extra heavy duty)
   • Type of prime mover (soft start vs. heavy start)
   • Hours of operation per day
   • For speed-increasing drives, multiply by an additional factor based on the speed ratio

2. Calculate design power:

   • Design power = Normal running power × Service factor
   • Normal running power = maximum power at the driver end (excludes shock loading and hard start factors)

3. Select the belt section using the belt selection chart (log-log plot of design power vs. rev/min of faster shaft):

   • Common sections: SPZ, SPA, SPB, SPC
   • When two or three sections may be suitable, check all possibilities for pulley size, number of belts, and cost before deciding
   • General rule: when two sections are suitable, choose the larger section (fewer belts needed with appropriately sized pulleys)

4. Determine the speed ratio:

   • Speed ratio = rev/min of faster shaft ÷ rev/min of slower shaft
   • If a speed tolerance applies, calculate the minimum and maximum driven shaft speeds
   • Maximum speed ratio obtainable with a single set of pulleys is approximately 6:1

5. Select the minimum pulley diameter from Table 1:

   • For intermediate values, choose the next closest larger size
   • If faster shaft speed > 2880 rev/min, the minimum pulley is that given for 2880 rev/min

6. Select suitable pulley pitch diameters:

   • Start with the minimum driver pulley diameter
   • Select approximately five of the next larger pitch diameters available
   • Multiply each driver pulley diameter by the speed ratio to determine the theoretical driven pulley diameter
   • Choose the closest available pitch diameter for the driven pulley
   • Calculate actual driven pulley speed = driver speed × (driver PCD / driven PCD)
   • Discard any combinations that give a driven speed outside the tolerance range

7. Choose the most suitable pulley combination:

   • Smaller pulleys → more compact drive, lower belt speed
   • Larger pulleys → higher power rating per belt, fewer belts required
   • Consider space constraints — maximum pulley sizes may be dictated by the available space

8. Calculate belt speed:

$$v = r \cdot \omega = \frac{d}{2} \times \frac{\pi \times N}{30}$$

• Where: d = pitch diameter (m), N = rev/min
• Belt speed must not exceed 40 m/s — if it does, select smaller pulleys

9. Estimate centre distance (if not specified):

   • A good design rule: centre distance ≈ sum of the two pulley pitch diameters (C ≈ D + d)

10. Calculate the required belt pitch length:

$$L = 2C + \frac{(D - d)^2}{4C} + \frac{\pi}{2}(D + d)$$

• Where: L = pitch length (mm), C = approximate centre distance (mm), D = large pulley pitch diameter (mm), d = small pulley pitch diameter (mm)

11. Choose a standard belt length closest to the calculated value from the belt section table:

• Generally choose the next larger rather than smaller size

12. Calculate exact centre distance:

$$C = A + \sqrt{A^2 - B}$$

• Where:

$$A = \frac{L}{4} - \frac{\pi}{8}(D + d)$$

$$B = \frac{(D - d)^2}{8}$$

• L = standard belt length chosen

13. Obtain basic power rating per belt:

• From the rated power tables for the chosen belt section
• Basic power per belt = Rated power + Additional power
• For speeds intermediate to those listed, use linear interpolation

14. Apply combined correction factor:

• From the combined arc of contact and belt length correction factor table
• Corrected power per belt = Basic power per belt × Combined correction factor

15. Calculate the number of belts:

• Number of belts = Design power ÷ Corrected power per belt
• Round to the nearest whole number
• Rule of thumb: if the fractional part is ≥ 0.3, round up; if < 0.3, round down (subject to design accuracy and safety factor assessment)

16. Obtain catalogue numbers for pulleys and bushes:

• Verify that pulleys are available with the required number of grooves
• Verify that the maximum bush bore does not exceed the shaft sizes being used
• If not available, it is usually preferable to redesign the drive rather than manufacture custom pulleys

Worked Example: Wedge Belt Drive Design

• Given: 6-cylinder diesel engine at 1050 rev/min driving a reciprocating gas compressor at 650 rev/min ± 3%; maximum power = 50 kW; operation > 16 hours/day; engine shaft = 70 mm diameter; compressor shaft = 80 mm diameter

• Step 1: Service factor = 1.4 (Class 3 heavy duty, internal combustion engine with ≥ 4 cylinders, over 16 hours/day)

• Step 2: Design power = 50 × 1.4 = 70 kW

• Step 3: Either SPB or SPC section is suitable; select the larger SPC section

• Step 4: Speed ratio = 1050/650 = 1.615 (reduction); driven speed range = 630.5 to 669.5 rev/min

• Step 5: Minimum pulley diameter = 250 mm (from Table 1)

• Step 6: Six combinations evaluated (driver pulleys from 250 to 335 mm); all give acceptable driven speeds within the tolerance range

• Step 7: Selected combination: driver = 315 mm PCD, driven = 500 mm PCD → driven speed = 661.5 rev/min; larger pulleys chosen for higher power rating

• Step 8: Belt speed = (0.315/2) × (π × 1050/30) = 17.3 m/s → OK (< 40 m/s)

• Step 9: Centre distance = 315 + 500 = 815 mm

• Step 10: Required belt pitch length = 2921 mm

• Step 11: Closest standard belt lengths: 2800 and 3150 mm → choose 3150 mm

• Step 12: Exact centre distance = 930.3 mm

• Step 13: Basic power per belt = 26.725 + 2.405 = 29.13 kW

• Step 14: Combined correction factor = 0.9 → Corrected power per belt = 29.13 × 0.9 = 26.217 kW

• Step 15: Number of belts = 70 / 26.217 = 2.67 → use 3 belts

• Step 16: Driver pulley: 315 mm PCD, 3 grooves, shaft 70 mm; Driven pulley: 500 mm PCD, 3 grooves, shaft 80 mm — both verified as available from catalogue

Service Factor Table (Summary)

Soft start: electric motors (star-delta, shunt wound, direct-on-line, series & compound wound), internal combustion engines with ≥ 4 cylinders, all prime movers fitted with centrifugal clutches, dry or fluid couplings, or electronic soft start devices

Heavy start: electric motors (direct-on-line, series & compound wound), internal combustion engines with < 4 cylinders, prime movers not fitted with soft start devices

For speed-increasing drives, multiply the service factor by an additional factor:

• Speed ratio 1.00–1.24 → multiply by 1.00
• Speed ratio 1.25–1.74 → multiply by 1.05
• Speed ratio 1.75–2.49 → multiply by 1.11
• Speed ratio 2.50–3.49 → multiply by 1.18
• Speed ratio 3.50 and over → multiply by 1.25

Belt Section Selection Guide (Table 2 Summary)

• Overlapping regions exist — when two sections may be suitable, evaluate both before selecting
• The selection chart is a log-log graph with rev/min of the faster shaft on the x-axis and design power on the y-axis

Combined Arc of Contact and Belt Length Correction Factor

• Correction factors are provided for each belt section (SPZ, SPA, SPB, SPC) based on:
  • Speed ratio (1–1.5, >1.5–2, >2–2.5, >2.5–3, >3)
  • Belt length (varies by section)
• Values range from 0.8 to 1.15 — shorter belts and higher speed ratios receive lower correction factors; longer belts and lower speed ratios receive higher factors
• A correction factor of 1.0 represents a nominal (baseline) condition

Power Rating Tables

• Rated power per belt is given for each belt section (SPZ, SPA, SPB, SPC) as a function of:
  • Rev/min of the faster shaft (100 to 6000 rev/min)
  • Small pulley pitch diameter (varies by section)
• Additional power per belt is also given for each speed ratio range (1.00–1.06 through to higher ratios)
• Total basic power per belt = Rated power + Additional power
• Belt speed range: only use pulleys from the specified manufacturer when belt speed falls between 30 and 40 m/s; confirm selection and supply with the manufacturer

Journal Bearings vs. Rolling Element Bearings

Belt Section Comparison

Lubrication Regime Comparison

Journal Bearing Selection Procedure

flowchart TD
    A[Obtain shaft diameter, speed, and radial load] --> B[Select bearing length from standard table<br>First trial: L/d = 1]
    B --> C[Calculate bearing pressure<br>p = F / d × L]
    C --> D[Calculate surface velocity<br>v = d/2000 × 2πN/60]
    D --> E{Is v ≤ 1 m/s?}
    E -- Yes --> F[Check p against<br>velocity vs. max pressure table]
    E -- No --> G[Check p against<br>pressure vs. speed chart]
    F --> H{Is p ≤ max allowable?}
    G --> H
    H -- No --> I[Increase bearing length L<br>and recalculate]
    I --> C
    H -- Yes --> J[Calculate p·v factor]
    J --> K{Is p·v ≤ 0.53?}
    K -- Yes --> L[No auxiliary lubrication needed]
    K -- No --> M[Auxiliary lubrication required]
    L --> N[Calculate bearing modulus<br>M = μv / p]
    M --> N
    N --> O{Is M ≥ 75?}
    O -- Yes --> P[Thick-film lubrication assured<br>Record catalogue number]
    O -- No --> Q[Consider increasing lubricant<br>viscosity or redesign]
    Q --> N

Wedge Belt Drive Design Procedure

flowchart TD
    A[Determine service factor<br>from duty class table] --> B[Calculate design power<br>= Running power × Service factor]
    B --> C[Select belt section<br>from selection chart]
    C --> D[Determine speed ratio<br>and driven speed tolerance]
    D --> E[Select minimum pulley<br>diameter from Table 1]
    E --> F[Select suitable pulley<br>pitch diameter combinations]
    F --> G[Check all combinations give<br>driven speed within tolerance]
    G --> H[Choose most suitable<br>pulley combination]
    H --> I[Calculate belt speed<br>v = rω]
    I --> J{Is v < 40 m/s?}
    J -- No --> K[Select smaller pulleys<br>and recalculate]
    K --> F
    J -- Yes --> L[Estimate or specify<br>centre distance]
    L --> M[Calculate required<br>belt pitch length]
    M --> N[Choose closest standard<br>belt length ≥ calculated]
    N --> O[Calculate exact<br>centre distance]
    O --> P[Obtain basic power per belt<br>= Rated + Additional power]
    P --> Q[Apply combined<br>correction factor]
    Q --> R[Calculate number of belts<br>= Design power ÷ Corrected power]
    R --> S[Round to nearest<br>whole number]
    S --> T[Obtain pulley and bush<br>catalogue numbers]
    T --> U{Are pulleys available<br>with required grooves?}
    U -- Yes --> V[Design complete]
    U -- No --> W[Redesign drive<br>with different pulleys]
    W --> F

Lubrication Regimes in Journal Bearings

flowchart LR
    A[Rest / Very Low Speed] --> B[Boundary Lubrication<br>Metal-to-metal contact<br>High wear]
    B --> C[Low to Moderate Speed]
    C --> D[Thin-Film Lubrication<br>Transition regime<br>Moderate wear]
    D --> E[High Speed]
    E --> F[Thick-Film Lubrication<br>No contact, no wear<br>Fluid friction only]

    style B fill:#ff6b6b,color:#000
    style D fill:#ffd93d,color:#000
    style F fill:#6bcb77,color:#000

Belt Drive Types Classification

graph TD
    A[Belt Drive Types] --> B[V-Section Belts]
    A --> C[Flat/Toothed Belts]

    B --> D[Vee Belts<br>Older design, still common]
    B --> E[Wedge Belts<br>Deeper profile, higher rating]
    B --> F[Banded Belts<br>Multiple belts joined at top]
    B --> G[Multi-Pull Belts<br>Similar to banded, different profile]
    B --> H[Link Belts<br>Urethane linked sections<br>Heat/oil/chemical resistant]
    B --> I[CRE Belts<br>Cogged edge, smaller pulleys<br>High-speed drives]

    C --> J[Synchronous Belts<br>Toothed profile, no slip<br>Lower load capacity]

    style A fill:#4a90d9,color:#fff
    style B fill:#5ba8c8,color:#fff
    style C fill:#5ba8c8,color:#fff

Key Terms Glossary

• Journal — the portion of a shaft that rotates within a bearing; often the same diameter as the shaft itself
• Bush — another name for a journal (plain) bearing, typically a cylindrical sleeve pressed into a housing
• Bearing Modulus (M) — ratio of (lubricant viscosity × surface velocity) to bearing pressure; used to predict lubrication regime
• Boundary Lubrication — a condition where the lubricant film is too thin to prevent metal-to-metal contact between the journal and the bearing
• Thick-Film Lubrication — a condition where the lubricant film is thick enough to fully separate the journal from the bearing, eliminating wear
• Projected Area — the product of the journal diameter and bearing length (d × L); used to calculate bearing pressure
• p·v Factor — the product of bearing pressure and surface velocity; if it exceeds a threshold value (typically 0.53 for porous bronze bearings), auxiliary lubrication is needed
• Embeddability — the ability of a soft bearing material to absorb foreign particles without damaging the journal surface
• Powder Metallurgy — a manufacturing process where metal powders are compacted and sintered to form solid components (used for porous bronze bearings)
• Porous Bronze — a self-lubricating bearing material made from sintered copper and tin powders, pre-impregnated with oil
• Dynamic Viscosity (μ) — a measure of a fluid's resistance to flow; measured in centipoise (cp) where 1 cp = 0.001 Pa·s
• Service Factor — a multiplier applied to normal running power to account for the severity of operating conditions in belt drive design
• Speed Ratio — the ratio of the faster shaft speed to the slower shaft speed in a belt drive
• Pitch Diameter (PCD) — the effective diameter of a pulley at which the belt runs; determines the actual speed ratio
• Design Power — the product of normal running power and the service factor; the effective power used for belt selection
• Combined Correction Factor — a factor applied to the basic power rating per belt to account for the arc of contact and belt length
• Arc of Contact — the angle of wrap of the belt around the smaller pulley; smaller arcs reduce the effective power transmission
• Taper Lock — a type of bush used to mount pulleys onto shafts; enables easy installation and removal without keyway damage
• Centre Distance — the distance between the centres of the driver and driven pulleys
• Fatigue Load Limit (Pᵤ) — the load below which rolling element bearing fatigue life is theoretically infinite
• Basic Dynamic Load Rating (C) — the constant radial load that a rolling element bearing can theoretically endure for 1 million revolutions
• Basic Static Load Rating (C₀) — the maximum radial load a non-rotating rolling element bearing can sustain without permanent deformation

Quick Revision

• Journal bearings are low-cost, quiet, high-speed capable, but limited in radial load capacity and require careful attention to shaft finish, material hardness, and lubrication
• Three lubrication regimes exist: boundary (high wear), thin-film/transition (moderate wear), and thick-film (no wear) — the design goal is to achieve thick-film lubrication
• Bearing modulus M = μv/p — if M > 75, thick-film lubrication is likely; if M < 75, redesign or increase viscosity
• p·v factor threshold = 0.53 — if exceeded, auxiliary lubrication is mandatory for porous bronze bearings
• L/d ratio for journal bearings should be in the range 0.5 to 1.5
• Running clearance rule-of-thumb: 1/1000 of journal diameter
• Wedge belt design starts with the service factor, then progresses through design power → belt section → speed ratio → pulley selection → belt length → correction factor → number of belts
• Belt speed must not exceed 40 m/s
• Maximum speed ratio with a single set of pulleys is approximately 6:1
• Centre distance rule-of-thumb: C ≈ D + d (sum of pulley pitch diameters)
• When two belt sections are suitable, choose the larger section for fewer belts
• When choosing a standard belt length, prefer the next larger rather than smaller size
• Rolling element bearings are characterised by dynamic and static load ratings, speed ratings, and fatigue load limits — selection depends on bore size, load, speed, and required life
• Spherical roller bearings include additional calculation factors (e, Y₁, Y₂, Y₀) for combined radial and axial loading

JOINTS, SPRINGS & MACHINE ELEMENTS

Springs, Bolted Joints, Welded Joints, Power Screws & Machine Elements

Overview

• Scope: Comprehensive reference covering rolled steel sections, helical springs, bolted joints, welded joints, power screws, and machine element design
• Purpose: Provides standard design data, formulas, worked examples, and selection procedures for common mechanical components
• Application: Useful for mechanical design, machine element selection, stress analysis, and joint design in engineering practice

Rolled Steel Sections

• Standardised structural steel profiles (parallel flange channels, equal angles, unequal angles) with tabulated dimensional and mechanical properties
• Properties include cross-sectional area, second moment of area, section modulus, radius of gyration, and torsion constant
• Used for structural and machine frame design where bending, shear, and buckling resistance are required

Helical Springs

• Elastic elements that store and release energy, classified by load type (tension, compression, torsion, bending)
• Governed by the relationship between force and deflection (spring constant k)
• Design involves selecting wire diameter, mean coil diameter, number of coils, free length, and verifying stress against allowable limits

Bolted Joints

• Mechanical fastening method using threaded fasteners loaded in tension, shear, or combined loading
• Design requires consideration of preload, safety factors, bolt material strength, and joint type (friction vs. bearing)
• Tightening torque and bolt selection are critical for joint integrity

Welded Joints

• Permanent joints formed by fusing materials together, analysed using butt weld and fillet weld methods
• Weld stress is calculated as force per unit length of weld (line method) or as conventional stress on the weld throat area
• Bending and torsion in welds require section property calculations of the weld group

Power Screws

• Threaded devices that convert rotary motion to linear motion (or vice versa)
• Key parameters include pitch, lead, thread form, friction angle, and helix angle
• Efficiency depends on friction coefficient, helix angle, and thread geometry

Machine Element Design

• Encompasses the design of knuckle joints, levers, couplings, and similar connections
• Requires stress analysis (tensile, shear, bending, bearing) and appropriate selection of proportions based on strength of materials

Rolled Steel Sections

Parallel Flange Channels (PFC)

• Designation format: depth (mm) followed by "PFC" (e.g., 380 PFC, 300 PFC)
• Key properties tabulated:
  • Mass per metre (kg/m)
  • Depth of section, flange width, flange thickness, web thickness
  • Root radius, depth between flanges
  • Gross cross-sectional area
  • Second moment of area (Ix, Iy) about both axes
  • Section modulus (Sx, Sy) — elastic
  • Radius of gyration (rx, ry)
  • Torsion constant (J) and warping constant
• Section capacity tables provide form factors, yield stress values, and load capacities about both axes for various steel grades

Equal Angles (EA)

• Designation format: leg size × leg size × thickness (e.g., 200 × 200 × 26 EA)
• Properties include:
  • Mass per metre, actual thickness
  • Gross area of cross-section
  • Coordinate of centroid (pn, pp)
  • Second moment of area about x-axis, y-axis, and principal axes
  • Section modulus, radius of gyration
  • Torsion constant (J)
• Note: The principal axes (n-n and p-p) are rotated at 45° relative to the geometric axes for equal angles

Unequal Angles (UA)

• Designation format: long leg × short leg × thickness (e.g., 150 × 100 × 12 UA)
• Additional properties compared to equal angles:
  • Centroid coordinates differ for each axis direction
  • Principal axis orientation angle (α) must be considered
  • Properties tabulated about both geometric and principal axes

Helical Springs

Classification in the supplied reference

Common Spring Materials

• Plain high carbon spring steel
• Alloy steel (including stainless steel)
• Spring brass, bronze, or monel metal
• Non-metal solids such as neoprene rubber
• Gases such as air or nitrogen (gas springs)

Spring Constant k

$$k = \frac{F}{x}$$

• F = total force (N) and x = total deflection (mm), OR
• F = change in force (N) and x = change in deflection (mm)
• Units: N/mm
• The force-deflection diagram is a straight line passing through the origin (zero load = zero deflection)

Pre-load

• Most springs are pre-loaded — they carry a certain load (or exert a force) when working deflection is zero
• x₁ = pre-load deflection
• x₂ = total (maximum) deflection
• x = working deflection (change in deflection) = x₂ − x₁
• F₁ = pre-load force
• F₂ = maximum force
• F = change in force = F₂ − F₁

Pre-load Worked Example

• Given: A valve spring exerts 200 N closed and 250 N open; working deflection = 8 mm
• Solution:
  • Change in force: F = 250 − 200 = 50 N
  • Spring constant: k = 50 / 8 = 6.25 N/mm
  • Pre-load deflection: x₁ = 200 / 6.25 = 32 mm
  • Total deflection: x₂ = 32 + 8 = 40 mm

Stock Spring Selection Procedure

1. Calculate the spring constant k = F / x
2. Determine the maximum force
3. Look up a catalogue for a spring matching k and maximum force requirements
4. Record outside diameter, wire diameter, free length, spring rate, and maximum deflection

Spring Design (Custom Springs)

Spring Index C

$$C = \frac{D}{d}$$

• D = mean diameter of spring (mm)
• d = wire diameter (mm)

Allowable Stress f_all

• Depends on wire material properties, wire diameter (smaller wire → higher allowable stress), and service conditions

• For a safety factor: maximum calculated stress should not exceed 85% of the value read from allowable stress curves

Calculated Stress f

• Helical springs are stressed in torsional shear + bending
• Torsion shear stress formula:

$$f = \frac{16T}{\pi d^3}$$

• Since torque T = F × D/2:

$$f = \frac{8FD}{\pi d^3}$$

• Including the Wahl factor K (accounts for combined torsional shear and bending):

$$f = \frac{8KFD}{\pi d^3} \quad \text{(Formula 3)}$$

• Or in terms of spring index C:

$$f = \frac{8KFC}{\pi d^2} \quad \text{(Formula 4)}$$

• Wahl Factor:

$$K = \frac{4C - 1}{4C - 4} + \frac{0.615}{C} \quad \text{(Formula 5)}$$

• Key notes:
  • Stress f is caused by load F (not change in load)
  • Spring stress is independent of the number of coils
  • Do not confuse K (Wahl factor) with k (spring constant)

Number of Coils

$$n = \frac{Gd}{8C^3 k} \quad \text{(Formula 6)}$$

• G = modulus of rigidity of the spring wire (typically 78.6 GPa for spring steel)
• The smaller the spring constant, the greater the number of coils needed
• A large deflection also means a small spring constant → more coils
• Stress is independent of the number of coils (same wire diameter, mean diameter → same stress regardless of coil count)

Initial Length (Free Length) of a Spring

$$L = Nd + x_2 \quad \text{(Formula 7)}$$

• Where N = total number of coils, d = wire diameter, x₂ = total deflection
• The loop/end lengths depend on the form of attachment provided for each end of the spring

Free Length for Compression Springs

$$L = Nd + x_2 (1 + C_a) \quad \text{(Formula 8)}$$

• C_a = clash allowance — the amount by which the design deflection is increased to eliminate the possibility of coil clash under load
• Typical clash allowance: 20% (0.2)
• However, a check of catalogue springs usually reveals that they have a clash allowance between 30% and 40%

Buckling of Compression Springs

• As free length increases in proportion to diameter, the spring becomes more slender and may buckle under load
• If L/D > 10, the spring will most likely buckle under any load or deflection
• If L/D < 10, the spring will most likely buckle under any load (deflection dependent — check using buckling ratio graph)
• Use the x₂/L ratio (maximum deflection to free length ratio) to determine if buckling is likely by reference to the buckling graph

Design Procedure Summary

1. Assume a spring index C using typical values table and obtain a trial value for the mean diameter D and wire diameter d
2. Determine the maximum allowable stress (if not given) using service condition tables and the allowable stress graph
3. Calculate the Wahl factor K
4. Calculate the stress in the spring using Formula 3 or Formula 4
5. Compare the calculated stress to the maximum allowable stress — if too high, trial a larger wire diameter and repeat; if too small, trial a smaller diameter
6. Determine the spring constant (spring rate) k
7. Determine the total number of coils n and hence the number of active coils N
8. Calculate the free length of the spring (compression: Formula 8; extension: Formula 7)
9. If buckling is likely, check whether some guidance or support is needed
10. Summarise the design preferably with a sketch showing all relevant data

Compression vs Extension Springs — Stock Catalogues

• Compression springs: Catalogue includes outside diameter, wire diameter, free length, spring rate (R in N/mm), solid height, and approximate number of coils
• Extension springs: Catalogue includes outside diameter, wire diameter, free length, initial tension (T₁), spring rate, and approximate extended length
• Catalogue numbering is based on imperial (inch) sizes (e.g., C0360-025-2000 = compression spring, 0.360 in OD, 0.025 in wire dia, 2.000 in free length)