Stress

Stress is internal force per unit area. Learn normal (tensile/compressive) stress, bearing stress, allowable stress and factor of safety, stress concentrations, and thin-walled pressure vessels.

Two rods carry the same 10 kN load — a thin one snaps, a thick one doesn't. The load is the same but the intensity inside the material differs. That intensity, force divided by area, is stress, and it's what materials actually 'feel'.

Comparing working stress with the material's strength, divided by a factor of safety, is the core check of mechanical design.

An analogy

Standing on snow in boots versus snowshoes: your weight is identical, but snowshoes spread it over more area — lower stress, so you don't sink.

Tension and compression

A pulling load stretches the material (tensile stress, +); a pushing load squashes it (compressive stress, −). 1 MPa = 1 N/mm², a convenient unit because drawings are in millimetres. Mild steel yields around 250 MPa; aluminium 6061-T6 around 275 MPa.

Factor of safety

Loads, material properties and manufacturing all vary. Designers divide strength by a factor of safety: about 1.5–2 for well-known static loads, 3–4 for uncertain or impact loads, 5+ for lifting gear and pressure equipment governed by codes.

Stress concentrations

Holes, notches, fillets and threads disturb the flow of stress and raise it locally. A small circular hole in a wide plate triples the peak stress (K_t ≈ 3). Generous fillet radii and smooth transitions are the cheapest way to make parts stronger — especially under fatigue.

Pressure vessels

Internal pressure in a thin cylinder creates hoop stress pD/2t — twice the longitudinal stress. That's why sausages and overpressured pipes split lengthwise. Spheres have pD/4t everywhere, which is why gas storage tanks are often spherical.

The stress tensor and yield criteria

At a point, stress has 6 independent components (3 normal, 3 shear). Rotating the axes gives principal stresses (Mohr's circle). Ductile metals are checked with von Mises: σ_vm = √(σ₁² − σ₁σ₂ + σ₂²) (2D) compared with S_y; Tresca uses the maximum shear stress. FEM output is usually von Mises stress.

Formulas

• Normal stress: σ = P / A
• Allowable stress: σ_allow = S_y / n
• Stress concentration: σ_max = K_t · σ_nom
• Thin cylinder: σ_hoop = pD / 2t, σ_long = pD / 4t

Worked example: Stress in a tie rod

Given: Ø20 mm steel rod; Tension 50 kN

Find: Tensile stress and factor of safety (S_y = 250 MPa).

• A = π × 20²/4 = 314.2 mm²
• σ = 50 000/314.2 = 159.2 MPa
• n = 250/159.2 = 1.57

Answer: σ ≈ 159 MPa, n ≈ 1.57

Worked example: Plate with a hole

Given: Plate 60 mm wide × 8 mm thick, Ø20 hole; P = 50 kN; K_t ≈ 2.2 (net-section basis)

Find: Peak stress at the hole.

• Net area = (60 − 20) × 8 = 320 mm²
• σ_nom = 50 000/320 = 156.3 MPa
• σ_max = 2.2 × 156.3 = 344 MPa

Answer: ≈ 344 MPa — local yielding for mild steel

Common mistakes

• Wrong: Using gross area when the section has holes. — Right: Use the net area through the holes, and apply K_t for peak stress.
• Wrong: Mixing N, kN, mm and m. — Right: Work in N and mm → MPa, or kN and m → kPa, consistently.

Applications

• Bolts & tie rods: Tensile stress area of threads is checked against proof strength.
• Pressure vessels & pipes: Wall thickness is chosen from hoop stress, with code safety factors (ASME, EN 13445).
• Lifting equipment: Hooks, shackles and slings use high safety factors and proof testing.

Key takeaways

• σ = P/A; MPa = N/mm².
• Allowable = strength / factor of safety.
• Holes and notches raise stress by K_t.
• Thin cylinders: hoop = pD/2t (the critical one).

Going further

• Mohr's circle: A graphical rotation of the stress state: centre (σx + σy)/2, radius √(((σx − σy)/2)² + τ²) gives principal and maximum shear stresses.
• Residual stresses: Welding, heat treatment and cold work lock stresses into parts; shot peening deliberately adds compressive residual stress to improve fatigue life.
• Limit-state design: Structural codes factor loads up (×1.35–1.5) and strengths down (γ_M) instead of a single safety factor — the modern approach in Eurocodes and LRFD.