Stress and strain equations describe how materials respond to forces and deformation. These relationships are fundamental in engineering design, allowing calculation of stress, strain, stiffness and structural behaviour.
Key equations include stress (σ = F/A), strain (ε = ΔL/L), Young’s modulus (E = σ/ε), bending stress and torsion relationships used in mechanical and structural analysis.
This page provides a complete reference of stress and strain formulas used in engineering, including bending, torsion and pressure vessel equations.
Strain represents the deformation of a material relative to its original length and is a dimensionless quantity used in stress analysis.
Note: For more detailed stress & strain
notes refer to webpage Stress & Strain
Strain = Change in length (dL)over original length (L)
e = dL / L
Stress = Force (F) divided by Area withstanding Force (A)
σ = F / A
Young's Modulus E = Stress ( s ) / Strain(e). This is a property of a material
E = s / e
These equations are used to calculate stress, deformation and material behaviour under load. Engineers apply them to beams, shafts and pressure vessels to ensure safe design.
General Formula for Bending
A beam with a moment of inertia I and with Young's modulus E will have a
bending stress f at a distance from the Neutral Axis (NA) y and the NA will bend
to a radius R ...in accordance with the following formula.
M / I = s / y = E / R
Maximum bending moment: M = W × L / 4
Maximum deflection: δ = W × L³ / (48 × E × I)
Maximum bending moment: M = W × L / 8
Maximum deflection: δ = 5 × W × L⁴ / (384 × E × I)
Maximum bending moment: M = W × L
Maximum deflection: δ = W × L³ / (3 × E × I)
Maximum bending moment: M = W × L² / 2
Maximum deflection: δ = W × L⁴ / (8 × E × I)
Maximum bending moment: M = W × L / 8
Poisson's Ratio = ν = (lateral strain / primary strain )
Shear Modulus G = Shear Stress /Shear Strain
G = τ / ε = E / (2 .( 1 + ν ))
General Formula for Torsion
A shaft subject to a torque T having a polar moment of inertia J and a shear Modulus G will have a shear stress q at a radius r and an angular deflection θ over a length L as calculated from the following formula.
T / J = G . θ / L = t / r
More detailed notes on torsion calculations are found at webpage Torsion
For a thin walled cylinder subject to internal pressure P the circumferential
stress = σc
This stress tends to stretch the cylinder along its length. This is also called the longitudinal stress.
σc = P . d / ( 4 . t )
For a thin walled cylinder subject to internal pressure P the tangential
stress = σc
This stress tends to increase the diameter). This is also called the hoop stress.
σt = P . d / ( 2 . t )
The above two formulae are only valid if the ratio of thickness to dia is less than 1:20
The equations for the stresses in thick walled cylinders are derived on web page Cylinders
r1 = internal radius
r2 =outer radius
p1 = internal pressure
p2 = external pressure
σ r = radial stress
σ t =tangential stress
Consider a cylinder with and internal diameter d 1, subject to an internal pressure p 1. The external diameter is d 2 which is subject to an external pressure p 2. The radial pressures at the surfaces are the same as the applied pressures therefore
σ r = A + B / r 2
σ t = A - B / r 2
The radial pressures at the surfaces are the same as the applied pressures therefore
- p1 = A + B / r 12
-p2 = A + B / r 22
The resulting general equations are known as Lame's Euqations and are shown as follows
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If the external pressure is zero this reduces to
If the internal pressure is zero this reduces to
Stress and strain calculations are used in structural design, mechanical engineering, materials selection and failure analysis. These equations help engineers determine whether components can safely withstand applied loads and deformation.
Further information can be found on related pages including beam theory, material properties, strength of materials and structural analysis.
Stress is the force applied per unit area of a material, calculated as σ = F/A.
Strain is the deformation of a material relative to its original length, calculated as ε = ΔL/L.
Young’s modulus is the ratio of stress to strain and represents material stiffness.
Stress is force per unit area, while strain is the resulting deformation.
They are used in structural design, mechanical engineering and materials analysis.
Summary: Stress and strain equations are fundamental to engineering analysis, allowing calculation of material behaviour under load in beams, shafts and pressure vessels.
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