Glossary term
Stress
The internal force per unit area that develops inside a material when external loads are applied.
Definition
quantityStress is the internal force per unit area that develops inside a material when it is subjected to external loads, displacements, or thermal effects.
Stress is the fundamental quantity used in solid mechanics and structural engineering to evaluate whether a material or component can safely carry applied loads. It describes how internal forces are distributed across a cross-section and governs failure modes such as yielding, fracture, buckling, and fatigue. Stress depends on the geometry of the body, the nature and direction of the applied loads, and the boundary conditions. Unlike force, stress is an intensive quantity: it is defined at a point and can vary continuously throughout a body.
Stress is one of the central quantities in engineering mechanics. When a body is loaded — by forces, moments, pressure, temperature changes, or imposed displacements — its internal structure responds by developing internal forces that resist deformation. Stress is the measure of how intense those internal forces are at any given point. It is defined as the limit of the internal force \Delta F acting on an infinitesimal area \Delta A as that area shrinks to a point:
The SI unit of stress is the pascal (Pa), defined as one newton per square meter (\text{N/m}^2). In engineering practice, components are routinely designed to stresses in the megapascal (\text{MPa} = 10^6 \, \text{Pa}) or gigapascal range. In North American practice, pounds per square inch (psi) and kilopounds per square inch (ksi) remain common.
Normal stress and shear stress
Stress is not a scalar: it requires specifying both the intensity of the internal force and the orientation of the surface on which it acts. When the internal force acts perpendicular to the cross-sectional surface, the result is normal stress, denoted \sigma. Normal stress can be tensile (\sigma > 0 by convention) or compressive (\sigma < 0). When the internal force acts parallel to the cross-sectional surface — producing a sliding tendency — the result is shear stress, denoted \tau.
For a simple axially loaded bar with cross-sectional area A carrying a force F:
For a shaft carrying torque T, the shear stress at radius r from the neutral axis is:
where J is the polar moment of inertia of the cross-section. In beams, bending produces a normal stress distribution that varies linearly across the depth:
where M is the bending moment, y is the distance from the neutral axis, and I is the second moment of area.
Engineering significance
Stress analysis is the foundation of structural and mechanical design. Engineers must ensure that stresses remain below allowable limits throughout the component’s service life, accounting for load variability, material scatter, manufacturing defects, and safety factors. The allowable stress depends on the failure mode: yielding (compared to yield strength), brittle fracture (compared to fracture toughness), fatigue failure (compared to endurance limit), or creep (at elevated temperatures).
The full description of stress at a point in a three-dimensional body requires the stress tensor. Local geometric discontinuities produce stress concentrations that amplify stresses above the nominal value. Temperature gradients and thermal constraints generate thermal stresses. Multiaxial stress states are evaluated against failure criteria such as the von Mises criterion. Each of these topics is treated in a dedicated entry.
Common mistakes
A common mistake is reporting one nominal stress without stating whether it is axial, bending, shear, principal, local peak, average, or equivalent stress. Another is comparing elastic stress directly to a material limit without accounting for stress concentration, load combination, temperature, fatigue, residual stress, manufacturing defects, or uncertainty. A strong stress review states load case, cross-section, coordinate system, sign convention, material data, failure mode, safety factor, and whether the value is nominal, local, linearized, or equivalent.