Stress and Strain Basics
Stress (σ) is force per unit area [Pa or MPa]. Normal stress acts perpendicular to a surface; shear stress (τ) acts parallel. Strain (ε) is the dimensionless ratio of deformation to original dimension. In the elastic range: σ = Eε (Hooke's Law), where E is Young's modulus.
| Material | Young's Modulus E (GPa) | Yield Strength σy (MPa) |
|---|---|---|
| Steel (mild) | 200 | 250 |
| Aluminum alloy | 70 | 270 |
| CFRP (0° ply) | 135 | — |
| Concrete | 25–30 | — (compressive: 20–40) |
Mohr's Circle
Mohr's circle is a graphical method to find principal stresses and maximum shear stress from a general stress state (σx, σy, τxy):
σ1, σ2 = (σx+σy)/2 ± √[((σx-σy)/2)² + τxy²]
von Mises Stress and Yield
von Mises stress is a scalar measure of equivalent stress in multiaxial loading: σvm = √[(σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)²] / √2. Yielding occurs when σvm ≥ σy.
Safety Factors
| Application | Typical Safety Factor |
|---|---|
| Aerospace (strict QC) | 1.5–2.0 |
| General machinery (static) | 2.0–3.0 |
| Civil structures (static) | 3.0–4.0 |
| Dynamic / impact loads | 4.0–8.0 |
Fatigue Design
Under cyclic loading, failure can occur well below the static yield strength. Use S-N curves and the Goodman diagram to account for mean stress effects.
How to Use
- Input material properties: Young's modulus (e.g., 200 GPa for steel), yield strength (e.g., 250 MPa), and Poisson's ratio (0.3)
- Define loading conditions: apply normal stress (tension/compression in Pa), shear stress (in Pa), and loading angle
- Select analysis mode: view normal/shear stress components, generate Mohr circle diagram, calculate von Mises equivalent stress, and apply safety factor (typically 1.5–2.0 for static loads)
- Review stress tensor output and fatigue curves if cyclic loading is selected
Worked Example
Aluminum 6061-T6 rectangular plate (E = 69 GPa, yield = 276 MPa) subjected to biaxial loading: σx = 120 MPa tension, σy = 40 MPa tension, τxy = 30 MPa shear stress. Mohr circle calculation yields principal stresses σ1 = 135.3 MPa, σ2 = 24.7 MPa. Von Mises stress: σv = √(135.3² + 24.7² − 135.3×24.7) = 124.8 MPa. With safety factor 1.5, allowable stress = 184 MPa; design is adequate. For 10⁶ cycle fatigue life, reduce allowable to ~100 MPa.
Practical Notes
- Mohr circle displays all stress states at 0–360° rotation; principal stresses occur where shear = 0
- Von Mises accounts for all stress components; essential for ductile metals; use maximum principal stress for brittle materials (cast iron, ceramics)
- Safety factors: 1.25–1.5 for well-known static loads; 2.0–3.0 for variable/impact loads and welded joints
- Fatigue strength at 10⁶ cycles is ~0.4–0.5 of static yield for ferrous metals; verify with S-N curves