Mechanical Design Data and Machine-Element Reference: Bolted Joints

Bolted Joints

Types of Bolted Joints

Types of Fasteners

• Set screws — held in location by a thrust collar or bearing
• Cap screws — head bears directly on the member
• Head bolts and nuts — many types including hexagon head, cup oval, square neck, hexagon socket head cap screws
• Studs — threaded at both ends

Bolt Material and Grade

• Standard metric bolt grades include property classes 4.6, 4.8, 5.8, 8.8, 10.9, and 12.9
• Reading the grade: First number × 100 = ultimate tensile strength (MPa); first × second × 10 = yield stress (MPa)
  • Example: Grade 8.8 → UTS = 800 MPa, Yield = 640 MPa
• Standard bolts are available in both imperial (inch) and metric sizes
• Thread forms: ISO metric (coarse pitch) with fine pitch available in both systems

Bolt Dimensions

• Nominal diameter D = outside diameter of the thread (equals the pitch diameter of a hypothetical zero-thread-depth bolt)
• Pitch diameter = diameter of the thread at the point where tooth and space widths are equal
• Root diameter d₁ = inside diameter of the thread (also known as minor diameter)
• Pitch p = axial distance between successive threads
• Lead L = distance advanced by nut (or screw) for 1 rotation (for single-start thread, L = p)

Stress Area

• The stress area (also known as the tensile area, A_t) is used for bolt strength calculations
• It is based on the mean of the root diameter and pitch diameter
• A_t = (π/4) × d₁² where d₁ is approximately equal to the mean of root and pitch diameters

Bolt Loading — Tension

Design Procedure for Direct Tensile Load

1. Determine the total required preload: F = S × L (Safety Factor × Applied Load)
2. Select bolt size and material from tables to find a bolt with tensile area giving preload stress within the yield stress
3. Select appropriate number of bolts: N = F / (Y × A_s) where Y = yield stress, A_s = stress area
4. Specify tightening torque from recommended assembly torque tables
5. Position bolts as near as possible to the line of direct tensile loading

Safety Factors for Bolted Joints

Bolt Loading — Shear

• Shear load is taken on the shank of the bolt (not the thread)
• Shear stress: f_s = F / A_s where A_s = shear area = (π/4) × d² (shank diameter)
• Maximum permissible shear stress (from Table 13) is typically 200 MPa for standard bolt grades assuming bolt is less than 16 mm diameter
• For precision high-tensile grade (class 8.8), shear capacity is significantly higher

Bolt Loading — Combined Tension and Shear

• When a bolt carries both tensile and shear loads, stresses must be combined
• Combined stress formula:

$$f_{max} = \sqrt{(f/2)^2 + f_s^2} + f/2$$

• Where f = direct tensile stress, f_s = shear stress
• The bolt must be satisfactory in both tension and shear separately as well as combined

Bolted Bracket in Bending

• Direct shear load per bolt: F₁ = F / (number of bolts)
• Bending moment about the bolt group centroid: M = F × y
• Force on each bolt due to bending: F₁ = M / (Σy²) where y = distance from centroid to each bolt centreline
• The most highly stressed bolts are those furthest from the pivot point (centroid)
• Resolve forces into vertical and horizontal components, then combine vectorially

Bolted Bracket in Torsion

• Direct shear load per bolt: F₁ = F / N
• Torsional moment about the centroid: T = F × e (e = eccentricity)
• Force on each bolt due to torsion: perpendicular to the radius from centroid to bolt
• The most highly stressed bolt is the one at the greatest radial distance from the centroid
• Combine direct and torsional forces vectorially

Flexible Gasket Joints

• For joints with gaskets (sealing liquids or gases under pressure):
  • Design pressure load on bolts: Q = A × P (area × pressure)
  • Total preload required: W = Q + F where F = preload to seat the gasket
  • Typically, W = Q × 1.1 (add 10% for gasket seating)
  • Select bolt type with proof load stress appropriate to the application

Friction Type Joints

• Bolts fitted in clearance holes — load transferred by friction between clamped surfaces
• Higher bolt tension → higher clamping force → higher friction resistance
• Bolt preload should be reduced by a factor of 0.806 when using flexible gasket joints

Tightening Methods

How a Bolted Joint Carries Load

• Tension: External load resisted by bolt pre-tension; the joined members are stiffer than the bolt so they compress much less than the bolt extends — pre-load force maintains clamping; external load adds only a small increment to bolt tension
• Shear: In friction joints, load is carried entirely by friction between clamped members; in bearing joints, the bolt shank bears against the hole wall
• External load should not exceed the preload — if it does, the joint separates and the bolt carries the full external load (dangerous for fatigue)

General Rules to Reduce Fatigue Failure

1. Tighten bolt effectively to ensure an induced tension or preload in excess of the maximum external load
2. Observe general rules should be followed to minimise possibility of fatigue failure of bolts under high alternating or fluctuating stresses

Welded Joints

Types of Welds

• Butt welds: Full penetration weld joining two plates edge-to-edge
• Fillet welds: Triangular cross-section weld joining two surfaces at approximately right angles

Butt Weld Assumptions

• Welding has been carried out by a competent trade welder in accordance with correct welding procedures for the material being welded
• A welding rod has been used that has a strength at least equal to the un-welded plate
• Weld runs for the full width of the plate and if long welds are to be made, it is preferable that they be intermittent rather than continuous

Weld as a Line Method

• The weld is designed as a separate component with stress area A = t × L
• Where L = length of weld, t = throat thickness
• Line stress f is defined as:

$$f = \frac{F}{L} \quad \text{(units: N/mm)}$$

• For an applied load F (any direction), the stress in the weld is:

$$f^s = \frac{f}{t} = \frac{F}{t \times L} \quad \text{(units: MPa)}$$

Conventional Design Method

• The weld is designed as a separate component with stress area A = t × L
• Where t = throat thickness of the weld
• Two methods may be used for design of fillet welds:
  1. Weld as a line — treats the weld as having no thickness; section modulus Z has units of mm²
  2. Conventional method — treats the weld as an area; section modulus Z has units of mm³

Fillet Weld Throat Thickness

• For a standard fillet weld, the angle of the weld is 45°
• Leg length s = size of fillet weld specified by the leg length
• Throat thickness t = s × 0.707 (= s × sin 45°)
• Preferred weld sizes (in mm): 2, 3, 4, 5, 6, 8, 10, 12, 16

Design of Fillet Welds

• Fillet weld should be on both sides wherever possible to minimise distortion and stress
• For greater strength: plates should use an E48xx rod (UTS of 410 MPa)
• For low carbon and mild steel plates: a commonly used electrode welding rod is the E41xx × UTS 410 or E3 × 410 MPa
• Under conditions of steady or static load, the allowable weld stress is often taken as 0.3 × UTS
• For dynamic or cyclic loads: an appropriate design (safety) factor should be applied

Allowable Weld Stress

• If the shear stress of the welding rod is not known, a rule-of-thumb is to use 75% of the tensile strength
• For static or gradually applied loads: allowable weld stress would be 0.3 × 410 = 123 MPa (for E41xx rod)
• For dynamic/cyclic loads: apply an appropriate safety factor

Bending Loads in Fillet Welds

• Direct loads only: the line method has no advantage
• However, when there are bending or torsion loads: the line method is very useful
• The weld as a line method is illustrated in worked examples

Bending Stress Formula (Weld as Line)

$$f = \frac{Z}{M} = \frac{I}{y \cdot W} \quad \text{where } Z \text{ is the section modulus with units mm²}$$

• If the weld is not treated as a line, the section modulus of area I or Z would vary with each different weld size — using the weld as a line avoids this

Torsion Loads in Fillet Welds

$$f = \frac{T}{J} \times r$$

• Where: r = radius (distance from centroid to outer fibre), f = polar second moment of area of the section with units mm³, T = torque (Nmm)

Locating the Centroid of a Weld Group

• Break the weld into component lengths
• Calculate the centroidal distance using the first moment of area approach:

$$y_c = \frac{A_1 y_1 + A_2 y_2 + \ldots}{A_1 + A_2 + \ldots}$$

• Where y is the vertical centroidal distance and A = weld length × 1 (for line method)

Section Modulus Formulas for Common Weld Configurations

• A comprehensive table of formulas exists for 12 standard weld configurations including:
  • Single line along one edge
  • Two parallel lines (top and bottom)
  • C-shapes, L-shapes, rectangular, and circular weld groups
  • Each providing formulas for Z (bending about x-axis) and J (polar moment for torsion)

Power Screws

Purpose

• Convert rotary motion to linear motion (or vice versa)
• Used in vices, clamps, jacks, presses, machine tool lead screws, and similar mechanisms

Terminology

• Pitch p = axial distance between successive threads
• Lead L = distance advanced by the screw (or nut) for 1 rotation; for single-start thread: L = p; for multi-start: L = n × p where n = number of starts
• Pitch diameter d = diameter of a theoretical thread with zero thread depth but has the same lead as the actual screw
• Root diameter d₁ = inside diameter of the thread (also known as minor diameter)
• Nominal diameter D = outside diameter of the thread
• Friction angle φ = angle whose tangent equals the coefficient of friction: tan φ = μ
• Helix angle θ = angle of the thread helix: tan θ = nπ / (π × d) = L / (π × d)

Thread Forms and Dimensions

• Several thread form variations exist: square, modified square, trapezoidal metric (ACME equivalent), and buttress
• Square thread has highest efficiency but is difficult to manufacture
• Trapezoidal metric thread (face angle of 15°, included angle of 30°) is the most commonly used
• Buttress thread: designed for heavy loads in one direction only

Thread Depth and Relationships

• For the trapezoidal thread: thread depth t = 0.5 p, pitch diameter d = D − 0.5p, minor diameter d₁ = D − 1.5 × D (approximately)
• For the buttress thread: similar relationships with a face angle of 5° and included angle of 30°

Screw Torque and Thrust

For a Square Thread (Downward Thrust)

• Raising load (Formula 4):

$$T = F \times \frac{d}{2} \times \tan(\theta + \phi')$$

• Lowering load (Formula 5):

$$T = F \times \frac{d}{2} \times \tan(\phi' - \theta)$$

• Where φ' = effective friction angle, θ = helix angle

For Non-Square Face Angle

• If the thread face is inclined at angle α (in place of φ), use φ' in these formulas where:

$$\tan \phi' = \frac{\cos \alpha}{\mu}$$

• The friction angle φ is given by: tan φ = μ

Coefficient of Friction

• Depends primarily on: surface finish quality, type and frequency of lubrication, and number of revolutions or cycles
• Lowest coefficient: accurately machined threads with good surface finish and operating for some period with good lubrication (oil of suitable viscosity) — can be as low as 0.1
• Average value (mean of two extremes): μ = 0.125
• Start-up or initial friction is higher; for start-up conditions, values should be increased by one-third (multiply by 4/3)

Thread-Pitch Diameter Relationship

• Unlike fastening screws, power screws do not have standardised metric pitch/diameter sizes
• A useful guide table relates pitch p to approximate nominal diameters

Efficiency of a Screw Thread

$$\eta = \frac{W}{W_0} = \frac{2\pi \times T}{F \times L}$$

• Where W = work input (rotational), W₀ = work output (linear), F = thrust load, T = applied torque, L = lead
• For 1 revolution of the thread: output = F × L; input = T × 2π

Self-Locking

• A screw is self-locking when the helix angle θ equals or is less than the friction angle φ: θ ≤ φ
• Overhauling occurs when θ > φ (the load would lower the screw without applied torque)
• Multi-start threads will not self-lock (because the helix angle is too large) — a two-start thread would have θ = 11.1° which is not self-locking at typical μ values
• Self-locking is an important safety feature in many applications such as lifting devices

Collar Friction

• Each of the three methods used for converting rotary motion to linear motion (collar, thrust bearing, or ball screw) has different friction characteristics
• If bedding is used: T = μ × F × r_m where r_m = mean radius of the thrust face
• For collar friction: Formula 8 applies: T = μ × F × r_m
• Where r_m = mean radius = (D₁ + D₂)/4 for a flat bearing surface

Stress Analysis of Power Screws

• Axial stress in the root: f = F / A₁ where A₁ = (π/4) × d₁² (tensile area based on root diameter)
• Shear stress in the thread: f_s = bh / (F × 1.5) where b = thread length in nut, h = thread height
• Bending stress in the thread: f_b = (I × b × h) / (F × 3) — the thread is treated as a short cantilever
• Bearing pressure in the thread: p_b = F / (I × b) where I = thread depth, b = thread engaged length
• Number of threads in nut: n = d/p (Formula 9)
• Thread length in nut: b = n × p (Formula 10)
• Bearing pressure: p_b = (I × b) / F (Formula 11)
• Maximum allowable bearing pressure depends on speed and lubrication

Buckling of Power Screws

• The effective length and radius of gyration must be used to check if the screw column will buckle
• k = a / d₁ (radius of gyration / stress diameter ratio)
• Relationship between effective length and k depends on the degree of end restraint:
  • Both ends rigidly held: L_e = 0.7 L
  • Both ends pin-jointed (equivalent to cantilever): L_e = 2 L
  • One end rigid, other free: L_e = 0.85 L
  • Flexible end supports (pin-joint equivalent): L_e = L
• Check for buckling using the practitioner's formula (for short/intermediate columns) or Euler's formula (for long columns)

the practitioner's Column Formula

$$F_{cr} = f_y \times A_1 \left[ 1 - \frac{4\pi^2 E}{(k/L_e)^2 \times f_y} \right]$$

• This formula applies for the critical buckling force; a safety factor should be applied (typically 5)
• For a maximum slenderness ratio of 100, the maximum length (or travel) is approximately 600 mm

Machine Elements — Knuckle Joints

Description

• A knuckle joint connects two rods that are in the same line of action
• Consists of an eye (fork end), a fork (clevis), and a pin
• Loads are typically tensile or compressive along the rod axis

Good Proportions (Based on Rod Diameter d)

Stress Analysis of a Knuckle Joint

Pin

• Bending stress:

$$f_{bending} = \frac{2 \times p \times a}{F}$$

• Where a = distance between supports (related to fork and eye widths), F = applied force
• Shear stress:

$$f_{shear} = \frac{a \times e}{F}$$

• The pin is in double shear (two shear planes)

Eye

• Bending stress = 2pa / F
• Shear stress = ae / F
• Fork tensile stress = 2(d − D) × a / F

Fork

• Bending stress = bq / F
• Shear stress = 2be / F
• Fork tensile stress = (d − D) × b / F

Notes on Knuckle Joint Design

• Joint proportions may be cast or fabricated — they may also be relatively small
• Joints may be case or fabricated; if they are relatively small, they may be cast
• If the knuckle joint is of standard proportions with the same strength material, the rods are integral with the eye and fork
• In the knuckle joint illustrated, there is no separate bearing and rotational or oscillating motion occurs between the pin and eye or pin and fork
• For practice, you may be asked to calculate stresses in the eye, fork, and pin — with all stresses compared to allowable

Machine Elements — Levers

Design Principles

• A lever transmits force using a fulcrum (pivot point)
• The level of mechanical advantage depends on the relative distances of force application points from the fulcrum
• Critical design factor is usually the bending stress at the fulcrum (maximum bending moment location)
• A U-beam (I-beam section) may be used; bending stress is usually the most critical stress
• Fulcrum, roller arms (rocker arms) may also be forged
• The design of a lever follows standard design procedures and is best illustrated by example

Lever Cross-Sections

• Typical cross-sections: rectangular, circular, I-section, T-section
• The lever can be cast or fabricated
• If the lever has an integral boss, bending stress may be a maximum just outside the boss

Design Considerations

• Often the lever transmits forces using knuckle joints or similar connections
• If the boss is designed with grease nipples or oil holes, the lever can be designed without a separate bearing
• A critical design factor is the bearing pressure at the fulcrum — particularly if the lever is required to perform a large number of operating cycles
• In some cases, rolling element bearings are used; in other cases, plain journal bearings are used, with the design practice to fit the boss with grease nipples so lubrication can be applied

Joint Types Comparison

Spring Type Comparison

Helical Spring Design Process

flowchart TD
    A[Start: Define Requirements] --> B[Determine F, x, and service conditions]
    B --> C[Assume Spring Index C from Table]
    C --> D[Calculate trial D and d]
    D --> E[Determine allowable stress f_all]
    E --> F[Calculate Wahl Factor K]
    F --> G[Calculate actual stress f]
    G --> H{f ≤ 0.85 × f_all?}
    H -->|No - too high| I[Increase wire diameter d]
    I --> D
    H -->|Yes| J[Calculate spring constant k]
    J --> K[Calculate number of coils n]
    K --> L[Calculate free length L]
    L --> M{Compression spring?}
    M -->|Yes| N{Check L/D ratio for buckling}
    N -->|L/D > 10| O[Add guidance/support or redesign]
    N -->|L/D ≤ 10| P[Check x₂/L ratio on buckling graph]
    M -->|No| Q[Check hook stress for extension]
    O --> R[Finalise Design and Summarise]
    P --> R
    Q --> R

Bolted Joint Design Process

flowchart TD
    A[Start: Define Load and Joint Type] --> B{Joint Type?}
    B -->|Tension| C[Calculate total preload F = S × L]
    B -->|Shear| D[Calculate shear load per bolt]
    B -->|Combined| E[Calculate both tension and shear]
    C --> F[Select bolt material and grade]
    D --> F
    E --> F
    F --> G[Determine tensile/shear area from tables]
    G --> H[Calculate stress and compare to allowable]
    H --> I{Stress OK?}
    I -->|No| J[Select larger bolt or higher grade]
    J --> G
    I -->|Yes| K[Determine number of bolts N]
    K --> L[Select tightening method and torque]
    L --> M[Position bolts per design rules]
    M --> N[Check for fatigue if cyclic loading]
    N --> O[Finalise Design]

Power Screw Design Process

flowchart TD
    A[Start: Define Load F and Travel] --> B[Select thread form]
    B --> C[Choose pitch p and diameter D from table]
    C --> D[Calculate helix angle θ]
    D --> E[Determine friction coefficient μ]
    E --> F[Calculate friction angle φ]
    F --> G{Self-locking required?}
    G -->|Yes| H{θ ≤ φ?}
    H -->|No| I[Reduce lead / use single start]
    I --> C
    H -->|Yes| J[Calculate raising torque]
    G -->|No| J
    J --> K[Calculate lowering torque]
    K --> L[Calculate efficiency η]
    L --> M[Check thread stresses]
    M --> N[Check bearing pressure]
    N --> O{Buckling concern?}
    O -->|Yes| P[Check slenderness ratio and critical load]
    O -->|No| Q[Finalise Design]
    P --> Q

Welded Joint Analysis Process

flowchart TD
    A["Start: Define Load Type and Magnitude"] --> B{"Weld Type?"}

    B -->|"Butt Weld"| C["Design as plate<br/>f = F / A"]
    B -->|"Fillet Weld"| D{"Loading Type?"}

    D -->|"Direct only"| E["f = F / (t * L)"]
    D -->|"Bending"| F["Use line method<br/>Calculate section modulus Z"]
    D -->|"Torsion"| G["Use line method<br/>Calculate polar moment J"]
    D -->|"Combined"| H["Calculate direct + bending/torsion stresses"]

    C --> I["Compare stress to allowable"]
    E --> I

    F --> J["f_bending = M / Z<br/>Then divide by t"]
    G --> K["f_torsion = T * r / J<br/>Then divide by t"]
    H --> L["Combine stresses vectorially"]

    J --> I
    K --> I
    L --> I

    I --> M{"Stress <= Allowable?"}

    M -->|"No"| N["Increase weld size or length"]
    N --> D

    M -->|"Yes"| O["Specify weld size, length, and electrode"]

Key Terms Glossary

Quick Revision

• Spring constant: k = F/x — the slope of the force-deflection line; units N/mm
• Spring stress formula: f = 8KFD / (πd³) or f = 8KFC / (πd²) — always use the Wahl factor K
• Wahl factor: K = (4C−1)/(4C−4) + 0.615/C — always greater than 1; accounts for curvature and direct shear
• Number of coils: n = Gd / (8C³k) — more coils = softer spring (lower k)
• Compression spring free length: L = Nd + x₂(1 + Cₐ) — include clash allowance (typically 20%)
• Buckling check: If L/D > 10, the spring will likely buckle — provide guidance or redesign
• Bolt grade reading: First digit × 100 = UTS; first × second × 10 = yield (e.g., 8.8 → 800/640 MPa)
• Bolt preload: F = Safety Factor × Design Load; select bolt so stress < yield stress
• Bolt shear: Use shank area (not thread area); shear stress = F/A
• Combined bolt stress: f_max = √((f/2)² + f_s²) + f/2
• Weld throat: t = 0.707 × s (leg length); preferred sizes: 2, 3, 4, 5, 6, 8, 10, 12, 16 mm
• Weld line stress: f = F/L (N/mm); convert to MPa by dividing by throat t
• Power screw torque (raising): T = F × (d/2) × tan(θ + φ')
• Power screw torque (lowering): T = F × (d/2) × tan(φ' − θ)
• Self-locking condition: θ ≤ φ (helix angle ≤ friction angle)
• Power screw efficiency: η = (F × L) / (2π × T)
• Knuckle joint proportions: Pin diameter = rod diameter d; eye OD = 2d; fork OD = 2d
• Fillet weld in bending: Use section modulus Z (mm² in line method) — select from table of standard configurations
• Fillet weld in torsion: Use polar moment J (mm³ in line method) — combine direct and torsional line stresses vectorially

Context and scope

A machine that fails does not announce itself with a press release. It announces itself with a scream — of metal, of operators, of accountants.

Every rotating shaft, every loaded joint, every belt-driven assembly in the world depends on a handful of fundamental components called machine elements. Bearings. Keys. Clutches. Seals. Belts. Chains. Motors. Get any one of them wrong, and you do not get a slightly worse machine. You get a catastrophe wrapped in a maintenance invoice.

And here is the uncomfortable truth most engineers discover too late:

THE PROBLEM: Machines Fail Because Designers Choose Components Like They Are Shopping for Groceries

Somewhere right now, a plant manager is staring at a seized bearing that was supposed to last five years. It lasted eleven months. A maintenance crew is tearing apart a gearbox because the wrong key was specified — a key that sheared under a shock load the designer never accounted for. A production line is silent because a coupling failed, and the replacement has a six-week lead time.

These are not freak accidents. They are design-stage decisions that went wrong because someone treated machine element selection as an afterthought.

The core problem breaks down into three failures:

• Wrong type selected. A plain bearing where a rolling-element bearing was needed. A rigid coupling where a flexible one was mandatory. A flat belt where a synchronous belt was essential.
• Wrong size calculated. Undersized for the load. Oversized for the speed. Mismatched for the thermal environment.
• Wrong operating conditions assumed. Lubrication that degrades at actual operating temperatures. Materials that corrode in the actual chemical environment. Seals that leak under the actual pressure differential.

The damage is not limited to the failed component. A single bearing failure can cascade into shaft damage, housing warping, seal destruction, and contamination of an entire lubrication system. One wrong key can shear and send a flywheel into an uncontrolled state. One improperly selected O-ring can cause a hydraulic system to lose pressure — and with it, the ability to stop a multi-ton press.