How to Calculate Stress and Strain
Learn how to calculate engineering stress, strain, and Young's modulus. Covers tensile, compressive, shear, and thermal stress with worked examples.
What Are Stress and Strain?
Stress (σ) is the internal force per unit area within a material resisting an applied load: σ = F / A, measured in Pascals (Pa) or MPa. Strain (ε) is the resulting deformation — specifically the change in length divided by the original length: ε = ΔL / L₀, a dimensionless ratio. Stress describes what the material experiences internally; strain describes how it actually deforms. Both are essential for predicting whether a component will survive its operating loads without permanent deformation or fracture.
Young's Modulus and Hooke's Law
Within the elastic region, stress and strain are linearly related by Young's modulus E (also called the elastic modulus): σ = E · ε. Rearranging, ε = σ / E and the elongation is ΔL = F · L₀ / (A · E). For structural steel, E ≈ 200 GPa; for aluminum, E ≈ 69 GPa; for concrete (compressive), E ≈ 25–35 GPa. A stiffer material (higher E) deforms less under the same stress. Hooke's law applies only up to the proportional limit — beyond it, the stress-strain relationship becomes nonlinear.
Types of Stress
Normal stress acts perpendicular to the cross-section and can be tensile (positive, pulling apart) or compressive (negative, pushing together). Shear stress (τ) acts parallel to the cross-section: τ = V / A for direct shear, where V is the shear force. Bearing stress is contact pressure between mating surfaces: σ_b = F / (projected contact area). In complex loading, combined stress states are analyzed using Mohr's circle or principal stress equations to find the maximum normal and shear stresses acting on any plane.
Poisson's Ratio and Lateral Strain
When a material is stretched axially, it contracts laterally. Poisson's ratio ν = −ε_lateral / ε_axial quantifies this behavior. For most metals, ν ≈ 0.25–0.35; for rubber, ν approaches 0.5 (nearly incompressible). Lateral strain: ε_lat = −ν · ε_axial = −ν · σ / E. This matters in thin-walled pressure vessels, biaxial loading scenarios, and press-fit calculations where lateral deformation affects interference. The three-dimensional generalized Hooke's law uses E and ν to relate all six stress and strain components.
Yield Strength and Factor of Safety
Yield strength (F_y or σ_y) is the stress at which a material begins to deform plastically. Design stress must stay below σ_y divided by a safety factor: σ_allow = σ_y / SF. Common safety factors: 1.5–2.0 for static ductile materials, 3–4 for brittle materials, 4–6 for fatigue loading. Ultimate tensile strength (UTS or σ_u) is the maximum stress before fracture. For ASTM A36 steel: F_y = 250 MPa, F_u = 400 MPa. Knowing the failure mode (yielding vs. fracture) guides selection of the appropriate allowable stress.
Shear Stress and Shear Modulus
Shear stress τ and shear strain γ are related by the shear modulus G: τ = G · γ. The shear modulus relates to Young's modulus through G = E / [2(1 + ν)]. For steel, G ≈ 79 GPa. Shear strain γ = Δx / h, where Δx is the lateral displacement and h is the element height. Torsional shear stress in a solid circular shaft is τ = T · r / J, where T is torque, r is the radial distance from the center, and J = π·d⁴/32 is the polar moment of inertia.
Stress Concentrations
Geometric discontinuities such as holes, notches, fillets, and keyways locally amplify stress by a concentration factor K_t: σ_max = K_t · σ_nominal. For a circular hole in a wide plate under uniaxial tension, K_t = 3. Sharp notches can produce K_t values of 5 or more. Stress concentrations are particularly dangerous under cyclic loading because fatigue cracks initiate at peak stress locations. Design mitigation includes increasing fillet radii, adding reinforcement around holes, and improving surface finish at critical sections.
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