Stress and Strain Fundamentals

FE Exam Weight: Strength of Materials accounts for 9-14 questions (~10% of the exam). This is a high-weight topic closely connected to Statics.

Types of Stress

Normal Stress (σ)

Force perpendicular to the cross-sectional area: $\sigma = \frac{P}{A}$

• Tensile stress (+): member is being pulled apart
• Compressive stress (-): member is being pushed together

Shear Stress (τ)

Force parallel to the cross-sectional area: $\tau = \frac{V}{A}$

Bearing Stress

Contact pressure between two surfaces: $\sigma_b = \frac{P}{A_b}$

where Ab is the projected area of contact (diameter × thickness for a pin in a plate).

Types of Strain

Normal Strain (ε)

$\epsilon = \frac{\Delta L}{L} = \frac{\delta}{L}$

Dimensionless (often expressed as in/in, mm/mm, or percentage).

Shear Strain (γ)

The angular deformation in radians: $\gamma = \frac{\Delta x}{L}$

Hooke's Law

In the elastic (linear) region:

$\sigma = E\epsilon \qquad \tau = G\gamma$

Relationships between elastic constants: $G = \frac{E}{2(1 + \nu)}$ $K = \frac{E}{3(1 - 2\nu)}$

where K = bulk modulus.

Axial Deformation

For a prismatic bar under axial load: $\delta = \frac{PL}{AE}$

For multiple segments or varying loads: $\delta = \sum_i \frac{P_i L_i}{A_i E_i}$

Thermal deformation: $\delta_T = \alpha L \Delta T$

where α is the coefficient of thermal expansion.

The Stress-Strain Diagram

Ductile vs. Brittle Materials

Factor of Safety

$FS = \frac{\text{Failure Stress}}{\text{Allowable Stress}} = \frac{\sigma_{failure}}{\sigma_{allow}}$

• For ductile materials: FS = σy / σallow (based on yield strength)
• For brittle materials: FS = σu / σallow (based on ultimate strength)
• Typical values: 1.5-3.0 for structural applications