Shear Stress Calculator
Enter a force and area for direct shear, a torque and shaft dimensions for torsional shear, or a shear force with beam geometry for transverse shear. The calculator returns the shear stress in your chosen units, the factor of safety when you supply a material allowable, and a worked step-by-step solution. Switch between metric and imperial at any time.
What is shear stress?
Shear stress is the internal force per unit area that acts parallel (tangential) to a cross-section, as opposed to normal stress, which acts perpendicular to it. When you slide a deck of cards sideways, or when a bolt holds two metal plates together under a lateral load, or when a shaft transmits a torque, the material resists by developing shear stress. The symbol is tau (Greek letter) and the SI unit is the pascal (Pa), with megapascals (MPa) and kilopascals (kPa) more common in practice. In the imperial system it is measured in pounds per square inch (psi). Three scenarios dominate engineering: direct (or punching) shear, where an applied force slides one surface past another; torsional shear, where a shaft is twisted by a torque; and transverse shear, which arises in beams loaded perpendicular to their axis.
Direct shear: tau = F / A
The simplest case is a force F acting uniformly across a shear plane of area A, giving tau = F / A. This covers bolts, rivets, pins, and keys where the fastener cross-section resists a transverse load. Real stress distributions are rarely perfectly uniform (there are stress concentrations around holes and under bolt heads), so engineers apply a safety factor, typically at least 2.0 for static loads and higher for fatigue or impact. For bolts in double shear, the resisting area is twice the bolt cross-section, halving the stress.
Torsional shear: tau = T r / J
When a shaft is twisted by a torque T, shear stress varies linearly from zero at the center to a maximum at the outer surface. The formula is tau = T * r / J, where r is the radial distance from the shaft axis and J is the polar moment of inertia. For a solid circular shaft of radius R, J = pi * R^4 / 2, so the surface stress is tau_max = 2T / (pi * R^3). For a hollow shaft with outer radius R_o and inner radius R_i, J = pi * (R_o^4 - R_i^4) / 2. Hollow shafts are more efficient than solid ones because the center material contributes little torque resistance but adds weight. The same formula applies to circular tubes, drive shafts, and any solid of revolution.
Transverse beam shear: tau = VQ / It
When a beam carries a load perpendicular to its axis, internal shear forces develop. The shear stress at any horizontal layer within the cross-section is tau = V * Q / (I * t), where V is the internal shear force, Q is the first moment of the area above (or below) the cut about the neutral axis, I is the second moment of area of the full cross-section, and t is the width of the section at that level. For a solid rectangle of width b and height h, the maximum shear occurs at the neutral axis and equals tau_max = 3V / (2bh) = 1.5 times the average shear stress (V / A). For a solid circular section, tau_max = 4V / (3A) = 1.33 times the average. In most long beams, bending stress is the critical failure mode, but shear governs in short, deep beams, near support points, and in thin webs of I-beams.
Typical shear yield strengths of common engineering materials
| Material | Tensile yield (MPa) | Shear yield ~ 0.577 * Sy (MPa) | Notes |
|---|---|---|---|
| Mild steel (A36) | 250 | 144 | Most structural steel sections |
| High-strength steel (A572 Gr 50) | 345 | 199 | Structural bolts, columns |
| Stainless steel 304 | 215 | 124 | Corrosion-resistant applications |
| 6061-T6 Aluminium | 276 | 159 | Lightweight structures, shafts |
| Aluminium 2024-T3 | 345 | 199 | Aerospace alloy |
| Grade 8.8 bolt | 640 | 370 | High-strength fasteners |
| Grade 12.9 bolt | 1100 | 635 | Ultra-high-strength fasteners |
| Brass (CW617N) | 140 | 81 | Fittings, valves |
| Copper (annealed) | 70 | 40 | Electrical connectors |
| Cast iron (grade 250) | n/a | ~170 (ultimate) | Brittle: use ultimate, not yield |
Shear yield strength is approximately 0.577 times the tensile yield strength (von Mises criterion). Values are approximate; always verify with material certifications.
Frequently asked questions
What is the difference between shear stress and normal stress?
Normal stress (sigma) acts perpendicular to a cross-section and results from axial loads (tension or compression) or bending. Shear stress (tau) acts parallel or tangential to the surface and results from transverse forces, torsion, or shear loads. On most structural elements both are present simultaneously, and failure theories such as von Mises or Tresca combine them into an equivalent stress for comparison against the material yield strength.
What units does this calculator use?
In metric mode, force inputs are in newtons (N), lengths in millimetres (mm), areas in mm squared, and the resulting shear stress is in megapascals (MPa), which equals N/mm squared. In imperial mode, forces are in pound-force (lbf), lengths in inches (in), and stress in pounds per square inch (psi). Note that torsion mode expects torque in N*m (metric) or lbf*in (imperial), and the calculator converts N*m to N*mm internally before computing.
How do I calculate shear stress in a bolt or pin?
Use direct shear mode. Enter the transverse force acting on the bolt or pin as the shear force, and the cross-sectional area of the bolt shank (pi * r squared) as the shear area. If the bolt is in double shear (shear planes on both sides), double the area. The result is the average shear stress across the bolt cross-section. Add an allowable shear stress (typically 60% of the material tensile yield strength) to get the factor of safety.
What is the factor of safety and what value should I use?
The factor of safety (FoS) is the ratio of the allowable (or yield) shear stress to the actual computed shear stress. A FoS of 1.0 means you are exactly at the limit; values above 1.0 provide margin. For static loads in well-understood, non-critical structures, FoS = 1.5 to 2.0 is common. For dynamic or fatigue loads, impact conditions, or safety-critical applications such as lifting equipment, FoS can be 3.0 to 5.0 or higher. Always check applicable codes (AISC, Eurocode, ASME) for mandated values.
Why does shear stress vary across a beam cross-section?
Because the shear flow must be zero at the free surfaces (top and bottom of a beam carry no stress). The stress builds up from zero at the outer edges to a maximum at the neutral axis. The VQ/It formula captures this by using Q, the first moment of the area above the cut: Q is largest at the mid-height of a rectangle (where the most area is concentrated above and below), making the stress highest there.
How does a hollow shaft compare to a solid shaft for torsion?
A hollow shaft with the same outer radius but with material removed near the center carries almost as much torque as a solid shaft while being considerably lighter, because the inner core contributes little to J. For example, removing 50% of the cross-sectional area from the center reduces J by only about 6% (the J of a hollow shaft with r_i = 0.5 r_o is J = pi/2 * (r_o^4 - (0.5 r_o)^4) = 0.9375 * pi * r_o^4 / 2). This is why drive shafts and structural tubes are usually hollow.
When does shear stress govern beam design instead of bending?
Shear governs when beams are short and deep relative to their span, when large point loads are applied close to supports, and in thin-webbed sections such as I-beams where the web is the primary shear carrier. As a rough rule, bending governs when the span-to-depth ratio exceeds about 5 to 10; shear is worth checking carefully when it is below that. Web buckling due to shear is a separate concern for slender webs.