Machine Design 2 - Formulario

Machine design 2 – compendium

Constraints

PVW ∫ δ . f . s = ∫ δ . X . A . dx where A = FX or N according to the loadForce method (+) N (+) P (+) R (+) δ (right) -(Φ . H)

Simple bending diagrams

Stress tensor

Mohr's circle and principal stresses

Principal stresses: det (X – Ф)

Hooke's law

ε_i = 1/E [σ_i - ν(σ_j + σ_k)]
γ_ij = τ_ij/G
G = E/2(1+ν)

Fatigue

σ_m = (σ_max + σ_min)/2
σ_a = (σ_max - σ_min)/2
Nlog N_f

Machine design 2 - compendium

Constraints

PVW1 - ∫_S = ∫_S xdA = ∫ xdX where A = FX/EJ on N/EJG according to the loadForce method + δ (right) -(Φ+P)

Simple bending diagrams

Stress tensor

Principal stresses:

Strain gauge rosette

ε_i = ε_x cos² θ_i + ε_y sin² θ_i + γ_xy cos θ_i sin θ_i

Hooke's law

ε_i = 1/E (σ_i - ν(σ_j + σ_k))
σ' = 2(I + D)/EG = E/2(I + D)

Stress concentration factor

σ_max = k·σ_nom
τ_max = kτ_nom

Polar coordinates

σ_r = σ_x + σ_y/2 + (σ_x - σ_y)/2 cos 2θ + τ_xy sin 2θ
τ_θ = - σ_x - σ_y/2 - (σ_x - σ_y)/2 cos 2θ - τ_xy sin 2θ

Fatigue

σ_m = σ_max + σ_min/2
R = σ_min/σ_max
σ_a = A·N_f^b

Design of components

If R=0 (σ_m=0)

σ_d=θ_e∙m_s∙m_mk_p

R=-1: Haigh diagram

Equivalent fully reversed stress σ_ae

σ_ae=σ_e ⇒ failure

σ_ar=σ_e^1-σ_m

Multiaxial fatigue

Bending + torsion

R=-1: Gough-Polland

R=-1 Sims

Variable amplitude loading

Palmer-Minera rule: D = Σ( m_i )

Fracture toughness

K_TC=βσ√πa

Precrack ön Slow crack growth

k_c = k_po = k_co

P_S=1/n ∑(1 / R - 1) ( P(Δ_K / Δ_KC))crack opening function

Nip

_M12 C= ^P/ ^s (1-M12)

Leak Before Break

Britto fracture

ΔP = HE = 2 (Δp _M - 2) C ΔP = HE

Fatigue of welds

Avoid crack at the root of filled weldsP: cross-section

Material specifications

Very hard steel: k ₇; 20000 cycles ~ 600 MPa

Moment-strain relationship

σ =NJ _{circular section} σ = 32Nπd³ ; hollow disk

Circular section σ=16Ne πd³ ; hollow disk σ = 60Nd⁴

HE beam

J web= ¹/ ₁₂ a (d-a²)

J flange= 2 . ¹/ ₁₂ b²

Transport movement

web(3): shear heads