Calculate shear stress from either a transverse load or torsional torque. Enter your applied force and cross-sectional area for direct shear, or input torque, shaft radius, and polar moment of inertia for torsional shear. Get back shear stress (τ) in Pascals along with supporting values for your structural or mechanical design analysis.
Shear Stress (τ)
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Shear Stress (kPa)
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Shear Stress (MPa)
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Shear Stress (psi)
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Formula Applied
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Shear stress is measured in Pascals (Pa) in the SI system, which equals N/m². In engineering practice you'll also see kPa, MPa, and GPa for larger values. In the imperial system, shear stress is expressed in pounds per square inch (psi) or ksi (kilo-psi).
Transverse shear stress (τ = F/A) results from a force acting perpendicular to the longitudinal axis of a member, causing sliding between cross-sections. Torsional shear stress (τ = T·r/J) arises when a torque or twisting moment is applied to a shaft, causing it to twist about its axis. Both produce shear stresses but act in different geometric configurations.
For a bolt or pin in single shear, the average shear stress is τ = F / A, where F is the applied force and A is the cross-sectional area of the bolt (A = π·d²/4). In double shear, the area is doubled because two cross-sections resist the load, so τ = F / (2A).
For a solid circular shaft of radius r, the polar moment of inertia is J = π·r⁴/2. For a hollow shaft with outer radius r_o and inner radius r_i, use J = π·(r_o⁴ − r_i⁴)/2. J has units of m⁴ in the SI system and in⁴ in the imperial system.
Maximum shear stress is the highest shear stress value within a cross-section. For a rectangular beam under transverse loading, the maximum shear stress occurs at the neutral axis and equals τ_max = 1.5 · V/A. For a circular shaft under torsion, the maximum shear stress occurs at the outermost surface (r = r_max), giving τ_max = T·r_max/J.
In Newtonian fluids, shear stress is directly proportional to shear rate through the dynamic viscosity: τ = μ · (du/dy), where μ is viscosity in Pa·s and du/dy is the velocity gradient perpendicular to flow. This relationship is known as Newton's Law of Viscosity. Non-Newtonian fluids do not follow this linear relationship.
For fully developed laminar flow in a cylindrical tube, the wall shear stress is τ_w = 4·μ·Q / (π·r³), where μ is dynamic viscosity, Q is volumetric flow rate, and r is the tube inner radius. This formula is commonly used in microfluidics and biomedical engineering to control the mechanical forces on cells.
The factor of safety (FOS) in shear is the ratio of the material's ultimate shear strength to the allowable (working) shear stress: FOS = τ_ultimate / τ_allowable. A higher FOS provides more safety margin against failure. Typical values range from 1.5 to 4 depending on the application, load certainty, and consequences of failure.