Stress and Strain Fundamentals

FE Exam Weight: Strength of Materials accounts for 9–14 questions (~10% of the 110-question FE Other Disciplines exam). It is open-resource — every equation below lives in the searchable NCEES FE Reference Handbook, so the exam tests whether you can locate the right formula and apply it with consistent units, not whether you memorized it.

Types of Stress

Normal stress (σ) is force acting perpendicular to a cross-sectional area:

σ = P/A

Tensile stress (positive) pulls a member apart; compressive stress (negative) pushes it together. Shear stress (τ) is force acting parallel to the area:

τ = V/A

Bearing stress is contact pressure, σ_b = P/A_b, where A_b is the projected contact area (for a pin of diameter d in a plate of thickness t, A_b = d·t — not the pin's circular area).

The unit trap

The single most common FE stress error is unit mismatch. In SI, work in newtons and millimeters because 1 N/mm² = 1 MPa exactly. So 500 mm² carrying 100,000 N gives 200 MPa directly. Mixing meters and millimeters in the same expression throws answers off by 10⁶.

Types of Strain

Normal strain (ε) is the dimensionless ratio of length change to original length:

ε = ΔL/L = δ/L

It is often reported in µε (microstrain, ×10⁻⁶) or percent. Shear strain (γ) is the angular distortion in radians, γ = Δx/L.

Hooke's Law and the Elastic Constants

In the linear-elastic region, stress is proportional to strain:

σ = Eε and τ = Gγ

The three constants are not independent. For an isotropic material:

G = E / [2(1 + ν)] and K = E / [3(1 − 2ν)]

where K is the bulk modulus. Notice that as ν → 0.5 (an incompressible material such as rubber), K → ∞ — the material resists volume change completely. Poisson's ratio ν = −ε_lateral/ε_axial: stretch a bar axially and it contracts laterally. A 0.001 axial strain with ν = 0.3 produces a −0.0003 transverse strain.

Axial Deformation

For a prismatic bar under axial load, elongation is:

δ = PL / (AE)

For stepped bars, varying loads, or different materials, sum each segment:

δ = Σ Pᵢ Lᵢ / (Aᵢ Eᵢ)

Worked example — axial elongation

A 2 m steel bar (E = 200 GPa, A = 400 mm²) carries an 80 kN tensile force. Find the elongation.

1. Keep N and mm: P = 80,000 N, L = 2,000 mm, A = 400 mm², E = 200,000 MPa.
2. δ = PL/(AE) = (80,000 × 2,000) / (400 × 200,000)
3. δ = 160,000,000 / 80,000,000 = 2.0 mm.

Check the stress: σ = P/A = 80,000/400 = 200 MPa, well below steel's ~250 MPa yield — the elastic formula is valid.

Thermal Stress and Strain

Free thermal expansion produces strain with no stress:

δ_T = α L ΔT

where α is the coefficient of thermal expansion (steel ≈ 12×10⁻⁶ /°C). But if the member is fully restrained between rigid supports, it cannot expand, so a thermal stress develops:

σ = E α ΔT (independent of length and area)

Worked example — restrained thermal stress

A steel bar fixed at both ends is heated 50°C. With E = 200 GPa and α = 12×10⁻⁶/°C:

σ = (200,000 MPa)(12×10⁻⁶)(50) = 120 MPa compressive.

The bar wants to grow but cannot, so it is compressed. Note the result is the same for a 1 m or a 10 m bar — length cancels.

The Stress–Strain Diagram

A tensile test traces the material's behavior from zero load to fracture:

Modulus of resilience is the elastic-region area under the curve (energy absorbed without permanent set); toughness is the total area to fracture. The slope of the initial straight line is E.

Ductile vs. brittle

Factor of Safety

FS = σ_failure / σ_allow

For ductile materials the failure stress is the yield strength σy (yielding is failure of function); for brittle materials it is the ultimate strength σu. Structural FS values typically run 1.5–3.0. A higher FS protects against overload, material variation, and analysis uncertainty but adds weight and cost.

Quick reference

• σ = P/A, τ = V/A, ε = δ/L
• δ = PL/(AE); thermal δ_T = αLΔT; restrained σ = EαΔT
• G = E/[2(1+ν)]; ν ≈ 0.3 for steel
• 1 MPa = 1 N/mm² — always work in N and mm in SI.