Home / Machine Elements

Machine Elements

Design of joints and connections: bolts, interference fits, parallel keys and welds.

Bolted Joint

preload and tightening torque (rough estimate)

Thread

Property class

Yield strength utilization ν [-] (typical 0.9)

Friction coefficient μ_tot [-] (oiled 0.12)

Tensile stress area A_s–

Preload F_V–

Tightening torque M_A–

Transmissible transverse force per interface (μ=0.15)–

Rough values from a simplified method. For critical joints: systematic calculation per VDI 2230 (German bolt standard).

Theory

During tightening, tensile and torsional stress act at the same time. The preload is chosen so that the equivalent stress reaches approx. 90 % of the yield strength:

σ_red = σ² + 3·τ² = ν · R_p0.2

Tightening torque:

M_A = F_V · ( 0.16·P + μ·( 0.58·d₂ + D_km2 ) )

P: thread pitch, d₂: pitch diameter
D_km: mean bearing diameter under the head
μ: friction coefficient in thread and under head (oiled ≈ 0.12)

Friction grip: F_Q = F_V · μ_T per interface.

Cylindrical Interference Fit

transmissible torque from interface pressure

Joint diameter d_F [mm]

Hub length L [mm]

Interface pressure p [N/mm²]

Static friction coefficient μ [-] (dry 0.15)

Safety factor S_r [-] (against slipping)

Operating torque M_t [Nm]

Joint area A_F–

Max. transmissible torque (slip torque)–

Allowable with safety factor S_r–

Assessment–

Transmissible axial force–

Theory

The interface pressure p presses shaft and hub together; friction transmits the torque:

M_max = p · μ · π · d_F² · L2

F_ax,max = p · μ · π · d_F · L

p results from the interference (Lamé equations)
μ: assembled dry ≈ 0.15–0.2, oiled ≈ 0.08–0.12
Check: M_t ≤ M_max / S_r with S_r ≈ 1.5–2

In addition, the hub must withstand the hoop stress (not covered by this calculator).

Parallel Key Connection

surface pressure per DIN 6885 (form A)

Shaft diameter d [mm]

Torque M_t [Nm]

Key height h [mm] (d=40: h=8)

Effective length l_tr [mm]

Number of keys [-]

p_allow [N/mm²] (steel hub ≈ 120)

Circumferential force F_u–

Surface pressure p–

Utilization–

Req. effective length–

With 2 keys, only ≈ 75 % is effective in practice (load factor 1.5 instead of 2). Round-ended form A: l_tr = l − b.

Theory

The governing value is the surface pressure on the hub keyway flank:

p = 2 · M_td · h′ · l_tr · n·φ ≤ p_allow

h′ ≈ 0.45·h: effective height of the keyway flank
l_tr: effective length (form A: l − b)
n·φ: effective number of keys (2 keys: φ = 0.75)
p_allow: cast iron hub ≈ 80, steel ≈ 120–150 N/mm²

Reduce p_allow accordingly for shock loading.

Fillet Weld under Tension/Shear

simplified stress check

Force F [kN]

Throat thickness a [mm]

Total weld length l_w [mm]

Allowable weld stress σ_w,allow [N/mm²] (S235 static ≈ 100–135)

Weld area A_w–

Weld stress σ_w–

Utilization–

Req. weld length at a–

Minimum throat thickness: a ≥ 3 mm and a ≥ √(t_max) − 0.5. For load-bearing structures, verification per EC3 (Eurocode 3) is required.

Theory

The stress is based on the design throat area of the weld:

σ_w = Fa · l_w ≤ σ_w,allow

a: throat thickness (height of the inscribed triangle)
l_w: effective weld length (excluding end craters)
shear loading is reduced by a factor of ≈ 0.65
σ_w,allow depends on material, weld quality and load case

Rough estimate for preliminary design. Does not replace a code-compliant design calculation. For critical joints, apply VDI 2230 (German bolt standard) or Eurocode 3.