Stress Calculator
Enter any two of the three variables - stress, force, or cross-sectional area - and the calculator solves for the third. Switch to the strain mode to add elongation data and compute Young's modulus, or pick a material preset to see how a common engineering material responds. All unit conversions happen automatically.
Formula
Worked example
A steel rod 20 mm in diameter is pulled by a 50 kN tensile force. Area A = π × (0.01 m)² ≈ 3.14 × 10⁻⁴ m². Stress σ = 50,000 N ÷ 3.14 × 10⁻⁴ m² ≈ 159.2 MPa. With E = 200 GPa, strain ε = 159.2 × 10⁶ Pa ÷ 200 × 10⁹ Pa ≈ 0.000796 (0.0796%). Safety factor against steel yield (250 MPa) = 250 ÷ 159.2 ≈ 1.57.
What is mechanical stress?
Mechanical stress (symbol sigma, or σ) is the internal force per unit area that develops inside a material when an external load is applied. It has units of pascals (Pa) or, more commonly in engineering, megapascals (MPa) or pounds per square inch (psi). Normal stress acts perpendicular to a cross-section (tensile when the material is being pulled apart, compressive when it is being squashed), while shear stress acts parallel to the cross-section. This calculator handles normal (axial) stress.
The stress formula and its three solve modes
The fundamental relationship is σ = F / A, where F is the applied force in Newtons and A is the cross-sectional area in square metres. Rearranging gives F = σ × A (solve for force) and A = F / σ (solve for area). Use the "Solve for" selector at the top to pick which variable you need, then fill in the other two. When you also know the strain (ε = change in length divided by original length), the calculator links in Hooke's Law, E = σ / ε, to find Young's modulus or solve for strain from a known modulus.
Strain and the elastic modulus
Strain is the fractional change in length: ε = ΔL / L, where ΔL is the elongation (positive for tension, negative for compression) and L is the original length. It is dimensionless. Young's modulus E connects stress and strain in the elastic region: σ = E × ε. A high modulus means the material is stiff - steel (200 GPa) stretches far less under load than rubber (0.05 GPa). This relationship holds only up to the proportionality limit; beyond it the stress-strain curve becomes non-linear and eventually the material yields.
Safety factor and yield strength
Engineers design structural members so that the working stress stays well below the material's yield strength. The safety factor is yield strength divided by applied stress; factors of 1.5 to 4 are typical depending on how well loads are known and how severe a failure would be. For example, structural steel with a yield strength of 250 MPa and an applied stress of 100 MPa gives a safety factor of 2.5. The reference table above lists approximate yield strengths alongside Young's modulus for common materials.
Young's Modulus of common engineering materials
| Material | Young's Modulus (GPa) | Typical yield strength (MPa) | Notes |
|---|---|---|---|
| Structural steel | 200 | 250 | Most common structural metal |
| Stainless steel (304) | 193 | 215 | Corrosion resistant, slightly less stiff |
| Aluminum alloy 6061-T6 | 69 | 276 | Lightweight, widely used in aerospace |
| Titanium (Grade 5) | 116 | 880 | High strength-to-weight ratio |
| Copper (annealed) | 110 | 70 | Highly ductile, good conductor |
| Concrete (typical) | 30 | n/a (brittle) | Strong in compression, weak in tension |
| Glass (soda-lime) | 70 | n/a (brittle) | Fractures rather than yields |
| Oak wood (along grain) | 12 | 100 | Anisotropic - very different across grain |
| Rubber (natural) | 0.05 | 15 | Very flexible, large strain before failure |
Approximate elastic moduli at room temperature. Actual values vary with alloy grade, heat treatment, and direction (for anisotropic materials like wood).
Frequently asked questions
What is the difference between stress and strain?
Stress (σ) is the internal force per unit area inside a material, measured in pascals or MPa. Strain (ε) is the resulting fractional deformation - how much the material stretches or compresses relative to its original length. Stress is the cause; strain is the effect. They are linked by Young's modulus: E = σ / ε.
What units should I use for stress?
In everyday engineering, megapascals (MPa) are the most practical unit because common structural materials yield at hundreds of MPa. One MPa equals one newton per square millimetre (N/mm²), which often maps cleanly to practical dimensions. The US still commonly uses psi or ksi (1 ksi = 1,000 psi ≈ 6.895 MPa). This calculator accepts Pa, kPa, MPa, GPa, psi, and ksi.
What is the difference between tensile and compressive stress?
Tensile stress occurs when a material is pulled apart (the cross-section is in tension, atoms are being separated). Compressive stress occurs when it is being pushed together (atoms are being compressed). Mathematically, compressive stress is negative by convention. Many materials - especially concrete and ceramics - are much stronger in compression than in tension, so the type of stress matters greatly for design.
What is Young's modulus and why does it matter?
Young's modulus (E) measures how stiff a material is: how much stress is needed to produce a given strain. A higher modulus means less deformation under load. Steel (200 GPa) is about three times stiffer than aluminum (69 GPa), which is why steel beams deflect less. However, two materials with the same modulus can have very different yield strengths - stiffness and strength are not the same thing.
When does Hooke's law break down?
Hooke's law (σ = Eε) is only valid in the linear-elastic region of the stress-strain curve, up to the proportionality limit. Beyond that point, most metals enter a non-linear elastic zone and then the yield point, after which deformation is permanent (plastic). Rubber and polymers behave non-linearly at much smaller stresses. Always check that your computed stress is below the yield strength of the chosen material before relying on these linear calculations.
How do I find the cross-sectional area for a circular rod?
For a solid circular cross-section of diameter d, the area is A = π × (d/2)² = π d² / 4. For example, a 20 mm diameter steel rod has A = π × 0.02² / 4 ≈ 3.14 × 10⁻⁴ m² = 314 mm². Enter that into the Area field. For a hollow tube, subtract the area of the inner circle from the outer: A = π(D² - d²) / 4, where D is the outer diameter and d is the inner diameter.