Stress
Quick Definition
Stress is the internal force per unit area in a loaded material: σ = F ÷ A. Normal stress pulls or pushes across a section, and shear stress slides along it. It is measured in pascals (N/m²), MPa or psi.
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Stress explained
A load tries to pull a part's atoms apart or slide them past each other. Stress measures how hard they are working, regardless of the part's size. Compare stress with what the material can bear: the first limit is yield strength, then tensile strength. Whether the margin is enough depends on the code and the load.
Synonyms & Related Terms
Formula
σ = F ÷ A
τ = F ÷ A (parallel to the area)
Real-World Example
A 25 mm rod carrying 50 kN has A = 490.9 mm², so σ = 50,000 ÷ 490.9 = 101.9 MPa. A steel with a 250 MPa yield has a factor of 2.45 over that. A 1 in rod carrying 10,000 lbf has A = 0.785 in², so σ = 12,732 psi (87.8 MPa).
Units of stress
Stress and pressure share a unit, but pressure acts on a surface from outside, while stress exists within the solid.
| Unit | Value |
|---|---|
| 1 Pa | 1 N/m² |
| 1 MPa | 1 N/mm² = 145.04 psi |
| 1 ksi | 1,000 psi = 6.895 MPa |
| 1 GPa | 1,000 MPa |
Four stresses you will calculate
Bending and torsional stress vary across a section and peak at the surface, so a thin hollow section can match a solid one. Pipes and tanks can be checked with the hoop stress calculator.
| Type | Formula | Example | Result |
|---|---|---|---|
| Axial | σ = F ÷ A | 25 mm rod, 50 kN | 101.9 MPa |
| Bending | σ = M × y ÷ I | 40 × 80 mm beam, 2 kN·m | 46.9 MPa |
| Hoop | σ = p × D ÷ (2t) | 10 bar, 200 mm, 5 mm wall | 20 MPa |
| Torsional | τ = T × r ÷ J | 40 mm shaft, 500 N·m | 39.8 MPa |
Stress and strain
Strain is the response: the stretch per unit length, with no unit. In the elastic range the two are proportional, σ = E × ε, where E is Young's modulus. A tensile test plots one against the other, and yield and tensile strength are read from that curve, as shown in What Is Stress and Strain? You can run this in the strain calculator.
Tests usually report engineering stress, based on the original area. True stress uses the current area, which shrinks as the bar stretches. At 300 MPa and 10% strain, true stress is 300 × 1.10 = 330 MPa. The two match at low strain, where design usually stays.
Combined stress
Real parts see several stresses together, so a single figure is needed to compare with a tensile test. For ductile metals, the usual figure is von Mises stress: σ_vm = √(σ² + 3τ²). For a normal stress of 120 MPa and a shear stress of 60 MPa, σ_vm = 158.7 MPa. The Tresca criterion, 2√((σ/2)² + τ²), gives 169.7 MPa and is slightly more cautious.
Where stress runs high
A section change, hole or thread concentrates stress. A small round hole in a wide plate raises the local stress to three times the nominal value, so 100 MPa becomes 300 MPa at the hole edge. Ductile metals yield locally and shrug this off under one static load, but fatigue cracks start here. Stress can also remain in a part with no load at all, after welding or heat treatment. This is called residual stress.
Why It Matters
Stress analysis is the foundation of structural and mechanical engineering. Every component — from a paper clip to a bridge cable — must be checked against stress limits. Underestimating stress leads to failures; overestimating forces oversized, heavy, and expensive designs. Getting stress right is the core engineering trade-off.
Industry Applications
Sources
- Budynas, R. G. and Nisbett, J. K., Shigley's Mechanical Engineering Design, McGraw-Hill
- International Organization for Standardization, ISO 6892-1, Metallic materials — Tensile testing
- Bureau International des Poids et Mesures, The International System of Units (SI Brochure)
Related Standards
Frequently Asked Questions
What is stress in engineering?
Internal force per unit area in a loaded material, σ = F ÷ A.
What are the units of stress?
Pascals, N/mm² (MPa), psi or ksi.
What is the difference between stress and strain?
Stress is the load intensity; strain is the deformation.
What is the difference between stress and pressure?
Pressure is applied to a surface; stress is inside a solid. Both use pascals.
What is shear stress?
Stress acting along a surface, as in a bolt under sideways load.
What is von Mises stress?
One equivalent value for combined stresses, used to check yield in ductile metals.
How do I convert MPa to psi?
Multiply by 145.04.
See Also
Related Glossary Terms
Yield Strength
Yield strength is the stress at which a material stops springing back and starts to deform permanently. For metals without a sharp yield point it is defined as the stress that leaves 0.2% permanent strain — the 0.2% proof stress, R_p0.2.
Tensile Strength
Tensile strength, or ultimate tensile strength (UTS), is the highest stress a material can carry in tension: the maximum force in a tensile test divided by the original cross-section. Ductile metals reach it before they neck and break.
Torsional Stress
Torsional stress is the shear stress a torque produces in a shaft. In a round section it rises from zero at the centre to a maximum at the surface: τ = T·r ÷ J, or τ = 16T ÷ πd³ for a solid shaft.
Elasticity
Elasticity is a material's ability to return to its original shape after a load is removed. In engineering, it is measured by a modulus such as Young's modulus. In economics, the same word means how strongly demand responds to price.
Shear Strain
Shear strain is the change in angle a material undergoes under shear, written γ and measured in radians. For small movements, γ = δ ÷ L. In the elastic range it is tied to stress by the shear modulus G, through τ = G × γ.
Microstrain
Microstrain (με) is strain expressed in parts per million: 1 με = 10⁻⁶ = 0.0001 %. It is the everyday unit for the small elastic strains measured in metals and structures, mostly with strain gauges.