Torsion Shear Stress Calculator
Calculate torsion shear stress accurately for your build. Get material quantities, waste allowances, and project cost breakdowns.
Formula
tau_max = T * c / J; theta = T * L / (G * J)
Where tau_max is the maximum shear stress, T is the applied torque, c is the outer radius, J is the polar moment of inertia, theta is the angle of twist, L is the shaft length, and G is the shear modulus of the material.
Worked Examples
Example 1: Solid Steel Drive Shaft
Problem: A solid steel shaft (D = 50mm, G = 80 GPa) transmits 500 N*m of torque over 1 meter. Find max shear stress and angle of twist.
Solution: J = pi x 50^4 / 32 = 613,592 mm^4\nMax shear stress: tau = (500 x 1000 x 25) / 613,592 = 20.37 MPa\nAngle of twist: theta = (500,000 x 1000) / (80,000 x 613,592) = 0.01019 rad = 0.5836 degrees
Result: Max shear stress: 20.37 MPa | Angle of twist: 0.584 degrees over 1 meter.
Example 2: Hollow Aluminum Tube
Problem: A hollow aluminum tube (D = 80mm, d = 60mm, G = 26 GPa) carries 200 N*m over 0.5m. Find stress and twist.
Solution: J = pi x (80^4 - 60^4) / 32 = pi x (40,960,000 - 12,960,000) / 32 = 2,749,018 mm^4\nMax shear stress: tau = (200,000 x 40) / 2,749,018 = 2.91 MPa\nAngle of twist: theta = (200,000 x 500) / (26,000 x 2,749,018) = 0.001399 rad = 0.0802 degrees
Result: Max shear stress: 2.91 MPa | Angle of twist: 0.080 degrees. Well within limits.
Frequently Asked Questions
What is torsion shear stress and how is it calculated?
Torsion shear stress is the internal stress developed in a shaft or structural member when a twisting moment (torque) is applied. The maximum shear stress occurs at the outermost fiber of the cross section and is calculated using the formula tau equals T times c divided by J, where T is the applied torque, c is the distance from the neutral axis to the outermost fiber (the outer radius), and J is the polar moment of inertia. For a solid circular shaft, J equals pi times the diameter to the fourth power divided by 32. The stress varies linearly from zero at the center to maximum at the surface. Understanding torsion shear stress is critical for designing drive shafts, axles, and structural connections.
How do hollow shafts compare to solid shafts for torsion?
Hollow shafts offer superior strength-to-weight ratios compared to solid shafts because the material near the center of a solid shaft carries very little shear stress. For example, a hollow shaft with an inner diameter equal to 80 percent of the outer diameter retains about 59 percent of the polar moment of inertia while using only 36 percent of the material. This means it can handle 59 percent of the torque at roughly one-third the weight. In practice, hollow shafts are widely used in automotive drive shafts, aircraft structures, and bicycle frames where weight savings are critical. The trade-off is that hollow shafts are more susceptible to local buckling under extreme loads.
What shear modulus values should I use for common materials?
The shear modulus G represents a material's resistance to shear deformation and is essential for calculating the angle of twist. Common values in gigapascals include: carbon steel at 79 to 84 GPa, stainless steel at 74 to 86 GPa, aluminum alloys at 25 to 28 GPa, copper alloys at 37 to 44 GPa, titanium alloys at 41 to 45 GPa, cast iron at 32 to 41 GPa, and brass at 35 to 40 GPa. For design purposes, using 80 GPa for steel and 26 GPa for aluminum are common approximations. The shear modulus is related to the elastic modulus E and Poisson ratio nu by the relationship G equals E divided by two times the quantity one plus nu.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
Is Torsion Shear Stress Calculator free to use?
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