Design for Static Loading and Failure Theories
Factor of safety, stress concentration, design of members under static loads using Tresca and von Mises theories, and the choice of theory for ductile and brittle materials.
Machine design asks: given the loads, what size must a part be so that it does not fail? For steady (static) loads the answer combines a failure theory with a factor of safety.
Factor of safety
N = failure stress / allowable (working) stress
- For ductile materials, failure stress = yield strength S_y.
- For brittle materials, failure stress = ultimate strength S_ut.
Higher N is used when loads are uncertain, materials vary, or failure would be dangerous.
Stress concentration
Holes, fillets, grooves and keyways raise the local stress: σmax = K_t σnominal, where K_t is the theoretical stress concentration factor.
- For brittle materials under static load, K_t must be applied.
- For ductile materials under static load, local yielding redistributes the stress, so K_t is usually ignored. Under fatigue loading it matters for both (see the next lesson).
Choosing a failure theory
With principal stresses σ₁ ≥ σ₂ ≥ σ₃:
| Theory | Design equation | Use |
|---|---|---|
| Maximum principal stress | σ₁ = S_ut/N | Brittle materials |
| Maximum shear stress (Tresca) | σ₁ − σ₃ = S_y/N | Ductile; conservative |
| Distortion energy (von Mises) | σ_vm = S_y/N | Ductile; most accurate |
Von Mises stress in 2D: σ_vm = √(σ₁² − σ₁σ₂ + σ₂²); for a shaft with bending stress σ and torsional shear τ: σ_vm = √(σ² + 3τ²), and Tresca gives √(σ² + 4τ²).
Shaft under combined loads
For a solid shaft of diameter d with moment M and torque T:
- Tresca: d³ = (32N/(πS_y)) √(M² + T²)
- Von Mises: d³ = (32N/(πS_y)) √(M² + 0.75T²)
Worked example
A point in a ductile part (S_y = 300 MPa) has σ = 120 MPa (bending) and τ = 40 MPa (torsion). Find the factor of safety by both theories.
- Tresca: √(120² + 4 × 40²) = √20800 = 144.2 MPa → N = 300/144.2 = 2.08.
- Von Mises: √(120² + 3 × 40²) = √19200 = 138.6 MPa → N = 300/138.6 = 2.17.
Von Mises always gives an equal or higher factor of safety than Tresca.
Practice
A plate with a hole has nominal stress 50 MPa and K_t = 2.5. What is the peak stress in MPa?
Show answer and explanation
Answer: 125σmax = K_t σnom = 2.5 × 50 = 125 MPa.
A point has σ = 80 MPa and τ = 60 MPa. What is the von Mises equivalent stress in MPa? (Give 1 decimal place.)
Show answer and explanation
Answer: 131.1σ_vm = √(80² + 3 × 60²) = √(6400 + 10800) = √17200 ≈ 131.1 MPa.
For a brittle material under static load, the appropriate failure theory is:
Show answer and explanation
Answer: ABrittle materials fail by separation when the largest normal stress reaches S_ut.
A ductile part has S_y = 240 MPa and a design factor of safety 3. What is the allowable stress in MPa?
Show answer and explanation
Answer: 80Allowable = S_y/N = 240/3 = 80 MPa.