Normal & Shear Stress

Calculate normal stress (σ = N/A) and shear stress (τ = F/A) for any cross-section.

Cross-sectional Area:

Inputs

N

Internal Force

A

Cross-sectional Area

mm²

Formula Interpretation

Normal Stress — Formula ①

\sigma = \dfrac{N}{A} \quad \text{(MPa)}

$N$ is the internal force perpendicular to the cross-section (N); $A$ is the cross-sectional area (mm²); $\sigma$ is the resulting normal stress (MPa).

Shear Stress — Formula ②

\tau = \dfrac{F}{A} \quad \text{(MPa)}

$F$ is the internal force parallel to the cross-section (N); $A$ is the cross-sectional area (mm²); $\tau$ is the resulting shear stress (MPa).

Circular Section Area

A = \dfrac{\pi}{4}\,d^2 \quad \text{(mm}^2\text{)}

$d$ is the diameter (mm); $A$ is the resulting cross-sectional area (mm²).

Knowledge Points

Normal Stress

The stress arising from internal forces perpendicular to a cross-section is called normal stress (σ). Tensile loading produces positive normal stress; compressive loading produces negative normal stress.

Shear Stress

The stress arising from internal forces parallel to a cross-section is called shear stress (τ). Shear stress causes angular (distortional) deformation of the material.

Five Types of Stress

There are five types of stress corresponding to five load types: tension/compression, shear, bending and torsion. Bending stress is a combination of tensile and compressive normal stress. Torsional stress is a form of shear stress. Stress is a tensor quantity.

Worked Example

As shown in the figure, a circular rod with diameter $30\,\text{mm}$ is subjected to tensile forces of $4000\,\text{N}$ at both ends. Find the tensile normal stress in the rod.

Step 1 — Identify the internal force

W = N = 4000\,\text{N}

Step 2 — Compute the cross-sectional area

A = \dfrac{\pi}{4}d^2 = \dfrac{\pi}{4} \times 30^2 = 225\pi \approx 706.86\,\text{mm}^2

Step 3 — Apply Formula ①

\sigma = \dfrac{N}{A} = \dfrac{4000}{225\pi} \approx 5.66\,\text{MPa}

The tensile normal stress in the rod is $\approx 5.66\,\text{MPa}$ .

Extended Knowledge

•In structural engineering, normal stress analysis is fundamental for designing columns, beams and cables under axial loading.
•Shear stress governs the design of bolts, pins, welds and adhesive joints that transmit loads through shear.
•When a member is simultaneously subjected to normal and shear stresses (combined loading), failure criteria such as von Mises or Tresca must be applied for safe design.
•For composite or variable cross-section members, stress distributions are non-uniform; advanced methods such as finite element analysis are required.
•Dynamic loads introduce stress concentrations at notches, holes and section changes. Fatigue analysis must apply stress concentration factors ( $K_t$ ) to predict service life.