Shear Stress Formula
The shear stress formula τ = F/A calculates the stress caused by forces acting parallel to a surface.
Learn with practical engineering examples.
The Formula
Shear stress is the stress that acts parallel (or tangential) to a surface, as opposed to normal stress which acts perpendicular to a surface. It occurs when forces try to slide one part of a material past another.
Think of cutting paper with scissors — the blades apply shear forces on opposite sides of the cut line. Shear stress is critical in engineering design because many structural failures occur due to shearing rather than simple tension or compression.
Understanding shear stress is essential for designing bolts, rivets, beams, and any connection where forces act sideways across a joint.
Variables
| Symbol | Meaning |
|---|---|
| τ | Shear stress (measured in pascals, Pa, or newtons per square meter, N/m²) |
| F | Shear force applied parallel to the surface (measured in newtons, N) |
| A | Cross-sectional area over which the force acts (measured in square meters, m²) |
Related Formulas
- Shear strain: γ = Δx / L (lateral displacement divided by original length)
- Shear modulus: G = τ / γ (ratio of shear stress to shear strain)
- Double shear: τ = F / (2A) (when a fastener is in double shear, the area doubles)
Example 1
A steel bolt with a diameter of 12 mm holds two plates together. A shear force of 25,000 N acts on the bolt in single shear. What is the shear stress?
Calculate the cross-sectional area: A = π(d/2)² = π(0.012/2)²
A = π × (0.006)² = π × 3.6 × 10⁻⁵ = 1.131 × 10⁻⁴ m²
Apply the formula: τ = F / A = 25,000 / 1.131 × 10⁻⁴
τ ≈ 221 MPa
Example 2
A rectangular aluminum key (6 mm × 6 mm cross-section) transmits torque in a shaft. The maximum allowable shear stress for the key material is 80 MPa. What is the maximum shear force the key can handle?
Calculate the shear area: A = 6 mm × 6 mm = 36 mm² = 36 × 10⁻⁶ m²
Rearrange: F = τ × A = 80 × 10⁶ × 36 × 10⁻⁶
F = 2,880 N (about 2.88 kN)
When to Use It
Use the shear stress formula in structural and mechanical engineering problems.
- Designing bolted and riveted connections
- Checking the shear capacity of beams and columns
- Analyzing punch and die cutting operations
- Calculating loads on adhesive joints
- Verifying the safety of pins and keys in mechanical assemblies