Formula & Calculator
Stress
Normal stress is force per unit area.
Interpretation
Stress is the internal resistance per unit area offered by a material to an external load. It is defined as force divided by the cross‑sectional area. Stress can be tensile (stretching), compressive (squashing), or shear (sliding). It is a fundamental concept in materials science and structural engineering.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| σ | Stress | Pa (N/m²) |
| F | Force | Newtons |
| A | Area | m² |
What it means
Stress (σ) is a measure of the intensity of internal forces acting within a deformable body. It is defined as the force per unit area, σ = F / A, where F is the applied force and A is the cross‑sectional area perpendicular to the force. Stress arises when external loads are applied to a material, causing it to deform. There are several types: normal stress (tensile or compressive) acts perpendicular to the surface, while shear stress acts parallel. Stress is a tensor quantity, but in simple cases it is treated as a scalar. The SI unit is the pascal (Pa), though megapascals (MPa) are common in engineering. Understanding stress is crucial for designing safe structures, as excessive stress can lead to yield or fracture. The concept is used in finite element analysis, failure theories, and material selection. Stress-strain curves provide essential material properties such as yield strength and ultimate tensile strength.
Worked example
| # | F (N) | A (m²) | σ (Pa) |
|---|---|---|---|
| 1 | 100 | 0.01 | 10000 |
| 2 | 250 | 0.02 | 12500 |
| 3 | 500 | 0.05 | 10000 |
| 4 | 1000 | 0.10 | 10000 |
| 5 | 150 | 0.03 | 5000 |
| 6 | 300 | 0.015 | 20000 |
| 7 | 75 | 0.005 | 15000 |
| 8 | 1200 | 0.08 | 15000 |
| 9 | 600 | 0.04 | 15000 |
| 10 | 200 | 0.025 | 8000 |
| 11 | 80 | 0.004 | 20000 |
| 12 | 450 | 0.03 | 15000 |
| 13 | 700 | 0.035 | 20000 |
| 14 | 900 | 0.06 | 15000 |
| 15 | 30 | 0.002 | 15000 |
Common mistakes
- Wrong area: Use the cross‑sectional area perpendicular to the load, not the total surface area.
- Unit consistency: Force in N, area in mm² gives N/mm² (MPa); for Pa use m².
- Sign of stress: Tensile is positive, compressive is negative – don’t ignore direction.
- Only for axial loads: This formula does not apply to shear or bending; use other formulas for those.
- Average stress: The formula gives average stress; local stress may vary if the section is not uniform.
Applications
Stress is a measure of the internal forces acting within a deformable body per unit area, arising from externally applied loads. It is a critical parameter in material science and structural engineering, as it dictates whether a component will deform elastically, plastically, or fracture under service conditions. Tensile stress occurs when forces pull apart, while compressive stress results from pushing together, and shear stress arises from tangential forces. Engineers use stress calculations to size cross-sections, select appropriate materials, and predict failure modes. For example, in bridge design, stress analysis ensures that cables and girders can support traffic loads without exceeding yield strength. In manufacturing, stress affects forming processes like rolling and forging. Understanding stress is essential for designing safe, reliable, and cost-effective structures and machines.
- Structural design of buildings, bridges, and towers
- Material selection for mechanical components (shafts, fasteners)
- Failure analysis and fracture prevention
- Finite element analysis (FEA) simulations
- Pressure vessel and piping design
Frequently Asked Questions
Stress is the internal resistance offered by a material to an externally applied load, defined as the force per unit area. It is a measure of the intensity of internal forces acting on a specific plane within the body. The basic formula is σ = F / A, where F is the force acting perpendicular to the cross‑section and A is the area over which it acts.
In the SI system, the unit is the pascal (Pa), which is 1 N/m². In practice, we use multiples:
- 1 kPa = 10³ Pa
- 1 MPa = 10⁶ Pa
- 1 GPa = 10⁹ Pa
- Using the wrong area – for hollow sections, you must use the net cross‑sectional area, i.e., subtract the inner area. For example, for a pipe, A = π/4 (D² – d²).
- Confusing force with stress – stress is force divided by area, not the force itself.
- Ignoring the sign convention – tension is positive, compression is negative. This is essential when combining stresses.
- Using the full cross‑section when holes or notches are present – stress concentration must be considered separately.
- Normal stress (σ) acts perpendicular (normal) to the cross‑sectional area and is caused by axial forces (tension or compression).
- Shear stress (τ) acts parallel to the cross‑sectional area and is caused by transverse forces or torsion.
In the linear elastic region, stress is directly proportional to strain (ε) via Hooke’s Law: σ = E · ε, where E is Young’s modulus (elastic modulus) of the material. Strain is the change in length per unit length (ε = ΔL/L). This relationship is the foundation of linear elasticity and is used to compute deformations and stiffness.
- Engineering stress is calculated using the original cross‑sectional area (A₀): σ_eng = F / A₀.
- True stress is calculated using the instantaneous area (A) that changes as the material deforms: σ_true = F / A.
The yield strength (σ_y) is the stress at which a material begins to deform plastically (permanently). It is the transition point from elastic to plastic behaviour. In design, the working stress must remain below the yield strength to avoid permanent deformation. The factor of safety is defined relative to yield strength for ductile materials.
The ultimate tensile strength (σ_UTS) is the maximum stress a material can withstand before fracture. It is the peak of the engineering stress‑strain curve. For ductile materials, this occurs after necking begins; for brittle materials, fracture happens at or near the ultimate stress. The UTS is used as the failure criterion for brittle materials and for overload calculations.
When the cross‑sectional area varies along the length, the stress is not uniform. You must calculate the stress at each section using the local area: σ(x) = F(x) / A(x). If the axial force is constant, the maximum stress occurs at the smallest area. For design, you typically check the smallest section (e.g., at a hole or a shoulder).
Stress concentration is the local increase in stress at geometric discontinuities such as holes, notches, keyways, and sharp corners. The stress concentration factor (K_t) is the ratio of the maximum local stress to the nominal stress: σ_max = K_t · σ_nominal. K_t depends on the geometry and is found from charts or finite element analysis. Designers reduce stress concentration by using fillets, rounded edges, and avoiding abrupt changes in cross‑section.