Mechanics of Materials: Stress & Strain

What is Stress?

Stress is the internal force per unit area within a material.

Normal Stress (σ)

σ = P / A

• P = axial force (tension + or compression -)
• A = cross-sectional area
• Units: Pa, MPa, ksi, psi

Shear Stress (τ)

τ = V / A (average shear)

• V = shear force
• More precisely: τ = VQ/Ib (shear formula)

What is Strain?

Strain is the deformation per unit length — dimensionless.

Normal Strain (ε)

ε = δ / L = ΔL / L

Shear Strain (γ)

Angular distortion in radians.

Hooke's Law

For elastic, linear materials:

σ = Eε

• E = Elastic (Young's) Modulus
• Steel: E = 29,000 ksi = 200 GPa
• Concrete: E ≈ 57,000√f'c (psi)
• Aluminum: E ≈ 10,000 ksi

Axial deformation: δ = PL/(AE)

Poisson's Ratio (ν)

Lateral strain / axial strain: ν = -ε_lateral / ε_axial

Typical values: 0.25–0.33 for metals

Shear Modulus

G = E / [2(1+ν)]

For steel: G ≈ 11,000 ksi

Shear Hooke's Law: τ = Gγ

Stress-Strain Curve (Steel)

1. Elastic region – linear, reversible
2. Yield point – F_y (steel yields)
3. Strain hardening – increased strength
4. Ultimate strength – F_u (maximum stress)
5. Fracture – failure

Key Values for A36 Steel:

• F_y = 36 ksi
• F_u = 58 ksi
• E = 29,000 ksi

Factor of Safety

FS = F_failure / F_allowable

Typically FS = 2–4 depending on application and load uncertainty.

Summary

Stress and strain are the language of structural behavior. Hooke's Law connects them in the elastic range. Know your material properties (E, F_y) and always check units.