Stress and strain in machine parts
Stress is the force on each unit of area inside a part: stress = F ÷ A. Units: pascal (Pa) or N/mm² (1 N/mm² = 1 MPa). A thin rod has more stress than a thick rod under the same load.
Strain is how much the part stretches compared with its length: strain = extension ÷ original length. It has no unit because it is a ratio.
Example: a 2 m wire that stretches by 1 mm has strain 0.001 ÷ 2 = 0.0005.
The stress–strain relation
For small loads, stress and strain go up together in a straight line. This is Hooke's law. The ratio is a number for each material: E = stress ÷ strain (Young's modulus). Steel has E about 200 GPa, so steel stretches very little.
As the load grows, a part passes through these points:
- Elastic region: it springs back when the load is removed.
- Yield point: after this it does not fully return.
- Plastic region: it stretches for good.
- Ultimate stress: the highest stress the part can carry.
- Fracture: it breaks.
Machine parts must always work in the elastic region.
Safety factor
Designers never load a part up to its breaking value. They keep a gap called the factor of safety: factor of safety = ultimate stress ÷ working stress. A factor of 4 means the part can carry four times the normal stress before it breaks. We use bigger factors where a failure can hurt people, like lifts and cranes, and because loads and materials are never perfect.
Shapes of machine parts
Shape matters as much as material. Stress piles up at sharp inside corners, small holes and deep scratches. This is called stress concentration, and cracks often start there. Good designs:
- Use a smooth curve (a fillet) where a shaft changes size.
- Change thickness slowly, not suddenly.
- Keep holes away from edges and round their edges.
- Use hollow tubes, I-sections and ribs to be strong with little weight, because the middle of a bar carries little stress in bending.
Try it: paper clip test
Take a paper clip and bend it a tiny bit. It springs back (elastic). Bend it far and it stays bent (plastic). Bend it back and forth and it snaps. Now cut a notch in the edge of a paper strip and pull; it tears at the notch. This is stress concentration, as in step 4 of the 3D.
Key formulas and definitions
- Stress σ = F ÷ A (1 N/mm² = 1 MPa)
- Strain ε = extension ÷ original length (no unit)
- E = stress ÷ strain (steel about 200 GPa)
- Extension = stress × length ÷ E
- Factor of safety = ultimate stress ÷ working stress
- Safe load = allowable stress × area
Worked examples
1. A rod of area 5 cm² carries 20 kN. Find the stress.
Area = 500 mm². Force = 20 000 N. Stress = 20 000 ÷ 500 = 40 MPa.
2. A 2 m wire stretches 1 mm. Find the strain.
Strain = 1 mm ÷ 2000 mm = 0.0005.
3. Stress is 100 MPa and strain is 0.0005. Find E.
E = 100 ÷ 0.0005 = 200 000 MPa = 200 GPa (steel).
4. A steel rod 3 m long has 100 MPa stress. E = 200 GPa. Find the extension.
Strain = 100 ÷ 200 000 = 0.0005. Extension = 0.0005 × 3000 mm = 1.5 mm.
5. A steel breaks at 400 MPa. The working stress is 100 MPa. Find the safety factor.
Factor = 400 ÷ 100 = 4.
6. A rod of 6 cm² may carry at most 80 MPa. Find the safe load.
Area = 600 mm². Load = 80 × 600 = 48 000 N = 48 kN.
Common mistakes
- Mixing units: 1 cm² is 100 mm², not 10 mm².
- Giving strain a unit. It is a ratio, so it has none.
- Thinking a thick rod is stronger only because of its material. For the same material, more area means less stress.
- Designing sharp inside corners. They start cracks; use a rounded fillet.