Stress Calculator

Normal stress (also called engineering stress or nominal stress) is defined as the force acting perpendicular to a surface divided by the area of that surface:

$$\sigma = \frac{F}{A}$$

where:

• σ (sigma) is the stress (Pa = N/m²)
• F is the applied force normal to the cross-section (N)
• A is the cross-sectional area over which the force acts (m²)

The SI unit of stress is the Pascal (Pa), which equals one Newton per square meter. Because the Pascal is a very small unit for engineering materials, stress is commonly expressed in:

• kPa (10³ Pa) — soil mechanics, low-pressure applications
• MPa (10⁶ Pa) — structural engineering, material properties
• GPa (10⁹ Pa) — elastic moduli, very high-stress applications
• psi (pounds per square inch) — US customary engineering

There are two fundamental types of normal stress:

Tensile stress occurs when forces pull the material apart (positive σ by convention). Examples include cables, ropes, and the bottom flange of a simply supported beam.

Compressive stress occurs when forces push the material together (negative σ by convention). Examples include columns, foundations, and the top flange of a simply supported beam.

The concept extends to more complex loading scenarios. Shear stress (τ = F/A) acts parallel to the surface. In general three-dimensional loading, the complete state of stress at a point is described by the Cauchy stress tensor — a 3×3 symmetric matrix with six independent components:

$$\boldsymbol{\sigma} = \begin{bmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{bmatrix}$$

Engineers compare calculated stress to a material's yield strength (σ_y) and ultimate tensile strength (σ_u) to assess safety. The factor of safety is FoS = σ_y / σ_applied. Typical design requires FoS of 1.5–4 depending on the application, consequences of failure, and load uncertainty.

The calculated stress should be compared against material strength limits. For structural steel (σ_y ≈ 250 MPa), stress below ~160 MPa provides a factor of safety around 1.5. If your stress exceeds the material's yield strength, plastic deformation will occur; if it exceeds the ultimate strength, fracture is imminent. Note this calculator assumes uniform stress distribution — stress concentrations near holes, notches, and fillets can multiply local stresses by factors of 2–5 or more.