Stress Formula
Calculate mechanical stress using σ = F/A.
Learn how force per unit area determines material stress in engineering applications.
The Formula
Stress measures the internal force per unit area within a material.
When an external force is applied to an object, stress tells you how that force is distributed across the cross-sectional area.
The SI unit of stress is the Pascal (Pa), which equals one Newton per square metre.
Variables
| Symbol | Meaning |
|---|---|
| σ | Stress (Pascals, Pa) |
| F | Applied force (Newtons, N) |
| A | Cross-sectional area (square metres, m²) |
Example 1
A steel rod with a cross-sectional area of 0.005 m² supports a 10,000 N load. What is the stress?
σ = F / A
σ = 10,000 N / 0.005 m²
σ = 2,000,000 Pa = 2 MPa
Example 2
A concrete column with an area of 0.25 m² carries a compressive load of 500,000 N. Find the stress.
σ = F / A
σ = 500,000 N / 0.25 m²
σ = 2,000,000 Pa = 2 MPa
When to Use It
Use the stress formula when you need to:
- Determine if a material can safely support a given load
- Compare the applied stress against a material's yield strength
- Design structural components like beams, columns, and rods
- Analyse tensile, compressive, or shear loading on a part
This formula applies to both tensile stress (pulling) and compressive stress (pushing).
For shear stress, the same formula applies but with the shear force and the area parallel to the force direction.
Key Notes
- Normal stress: σ = F/A: Force perpendicular to the cross-sectional area. Tensile stress (pulling apart) is positive; compressive stress (pushing together) is negative by convention. Units: Pa, MPa, or psi.
- Shear stress: τ = F/A: Force parallel to (along the surface of) the cross-sectional area. Shear failure is how bolts break, how scissors cut, and how landslides occur. Shear strength is typically 50–60% of tensile strength for metals.
- Yield strength vs ultimate tensile strength: Yield strength is where permanent deformation begins (the design limit). Ultimate tensile strength (UTS) is the maximum stress before fracture. Factor of safety = yield strength / working stress; typical values are 1.5–4 depending on application criticality.
- Stress concentration: Geometric discontinuities (holes, notches, sharp corners) amplify local stress. The stress concentration factor Kt relates peak stress to nominal stress: σ_max = Kt × σ_nominal. For a circular hole in a wide plate, Kt = 3 — the local stress is 3× the far-field stress.
- Applications: Stress analysis is used to size structural members, select materials, design bolted and welded joints, assess fatigue life, and verify that pressure vessels and pipelines remain below yield limits under all expected loading conditions.