Normal & Shear Stress
Compute normal stress σ (perpendicular force) and shear stress τ (parallel force) on a cross-section; direct area input supported.
Inputs
Formula
σ = F/A; τ = Fs/As
Fundamentals
Normal and shear stresses on a stress element together describe the stress state at a point — the basis of strength analysis.
Mutually perpendicular faces carry normal stresses $\sigma_x,\sigma_y$ and shear $\tau_{xy}$; complementary shear $\tau_{xy}=\tau_{yx}$. Principal planes have zero shear.
History
Stress was defined by Cauchy (1822); the stress state at a point and Mohr's circle were given by Mohr (1882).
Engineering applications
Used to analyse the stress state at a point, e.g. pressure vessels and shaft shoulders, and check strength.
Glossary
| Normal $\sigma$ | Stress normal to a plane, tension positive. |
| Shear $\tau$ | Stress parallel to a plane. |
| Complementary shear | Equal shear on orthogonal faces. |
How to use
- Fill in Axial force F, Area A, Shear force Fs, Shear area As in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Normal stress σ, Shear stress τ in the results area.
Formula notesPlane-stress principal σ1,2 = (σx+σy)/2 ± √[((σx−σy)/2)² + τ²]; max shear τ_max = √[((σx−σy)/2)² + τ²].
Formula · Worked Example · Knowledge
Formula
Normal stress $\sigma=F/A$ (force ⟂ area), shear $\tau=F_s/A_s$ (force ∥ area); MPa = N/mm².
Worked Example
F=10000 N, A=200 mm² → σ=50 MPa; Fs=5000, As=100 → τ=50 MPa.
Key Points
- Normal stress gives tension/compression; shear gives slip.
- Complex states use a stress element.
- Allowable stress follows material & safety factor.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Axial force F (N) | F | N | 10000 |
| Area A (mm²) | A | mm² | 200 |
| Shear force Fs (N) | Fs | N | 5000 |
| Shear area As (mm²) | As | mm² | 100 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Normal stress σ | sigma | MPa |
| Shear stress τ | tau | MPa |
Applications
- Common engineering use cases
FAQ
What formula does this tool use?
This tool computes per ISO / AGMA / ASME standard formulas: σ = F/A; τ = Fs/As
How accurate are the results?
Results match input precision, based on SI units and common engineering approximations; for critical duty re-check with a safety factor.
Where is it used?
Common engineering use cases