Compute average and maximum transverse shear stress for rectangular and circular cross-sections. Visualize parabolic shear stress distribution, ideal for mechanical design, beam analysis, and coursework.
Shear stress arises when a force is applied parallel or tangential to a material’s surface. In beams, transverse shear stress varies across the cross-section and is critical for assessing failure modes like shear rupture or excessive deflection. The general formula derived by Jean-Claude Saint-Venant and expanded by Euler-Bernoulli beam theory gives the shear stress at a specific fiber as τ = VQ/(Ib), where V is internal shear force, Q is first moment of area, I is moment of inertia, and b is width at that fiber.
For rectangular sections: τmax = (3/2)·(V/A) | For solid circular sections: τmax = (4/3)·(V/A)
Parabolic distribution: zero at outermost fibers and maximum at neutral axis.
The study of shear stress matured during the industrial revolution when railway axles and iron bridges failed unexpectedly. In 1856, Johann Bauschinger experimentally confirmed the parabolic shear distribution. Modern design codes (ASME, Eurocode 3) incorporate maximum shear stress criteria to prevent ductile failure. Understanding τmax guides pin sizing, bolted joints, and wooden beam design where shear often governs.
A rectangular lifting lug (b = 40 mm, h = 120 mm) sustains a transverse force of 25 kN from a hoist. Our calculator yields τ_avg = 5.208 MPa, τ_max = 7.812 MPa. The material (A36 steel) has allowable shear stress ≈ 100 MPa, ensuring safe operation. Designers often check both flexural and shear stresses, but thick components may be shear-critical.
Similarly, a circular pivot pin (d = 20 mm, V = 15 kN) experiences τ_avg ≈ 47.75 MPa, τ_max ≈ 63.66 MPa, which should be compared against yield strength in shear (typically 0.577·σ_y).
| Material | Typical τ_allow (MPa) | Notes |
|---|---|---|
| Structural Steel (A36) | 100 | Based on 0.4·σ_y, safety factor ~1.5–2.0 |
| Aluminium 6061-T6 | 55 | Shear yield ≈ 0.55·σ_y |
| Pine (wood, parallel to grain) | 6–8 | Varies with grade and moisture |
| Brass C26000 | 70 | Annealed condition |
⚠️ Always consult material data sheets and relevant design codes (ASME, Eurocode, etc.) for final design. The values above are for quick reference only.
Our implementation aligns with “Mechanics of Materials” by Beer, Johnston, DeWolf (9th Ed.) and ASME BTH-1 design guidelines. The shear correction factor for rectangular sections (1.5) and circular (1.333) is universally accepted in engineering mechanics. For further reading, refer to Engineering Toolbox and academic resources by MIT OpenCourseWare.