Torsion Calculator
Plan your structural engineering project with our free torsion calculator. Get precise measurements, material lists, and budgets.
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Adjust values & calculateTorsion Properties
Formula
Maximum shear stress equals the applied torque times the outer radius divided by the polar moment of inertia. The angle of twist equals the torque times the length divided by the product of shear modulus and polar moment of inertia. For solid circular sections, J equals pi times the radius to the fourth power divided by 2.
Last reviewed: December 2025
Worked Examples
Example 1: Solid Steel Shaft
Example 2: Hollow Shaft Comparison
Background & Theory
The Torsion Calculator applies the following established principles and formulas. Structural and construction engineering is governed by fundamental load analysis, material science, and regulatory standards that ensure the safety and durability of built structures. The primary distinction in load analysis is between dead loads — the permanent self-weight of structural elements, finishes, and fixed equipment — and live loads, which represent variable occupancy, furniture, and environmental forces such as wind and snow. These are combined using factored load equations, such as the ASCE 7 formula U = 1.2D + 1.6L, where D is dead load and L is live load. Concrete mix design is governed by the water-cement (w/c) ratio, which is the primary determinant of compressive strength and durability. A w/c ratio of 0.40–0.45 typically yields concrete with 28-day compressive strengths of 30–40 MPa. Common mix ratios by weight for structural concrete are approximately 1 part cement : 1.5–2 parts sand : 3 parts coarse aggregate. Structural steel is characterized by its yield strength (the stress at which permanent deformation begins, typically 250–350 MPa for mild steel) and ultimate tensile strength (typically 400–500 MPa). Mid-span deflection of a simply supported beam under a central point load is given by δ = FL³ / (48EI), where F is force, L is span length, E is Young's modulus, and I is the second moment of area. Building insulation is rated by R-value, a measure of thermal resistance in units of m²·K/W (SI) or ft²·°F·h/BTU (imperial). Higher R-values indicate greater resistance to heat flow. Foundation design depends on the allowable bearing capacity of the underlying soil, which ranges from approximately 75 kPa for soft clay to over 10,000 kPa for bedrock. Drainage gradients for surface water are typically specified as a minimum of 1–2% slope away from building foundations to prevent hydrostatic pressure and water infiltration.
History
The history behind the Torsion Calculator traces back through the following developments. The history of construction engineering spans thousands of years of accumulated empirical knowledge and, more recently, rigorous scientific analysis. The ancient Egyptians built the Great Pyramid of Giza around 2560 BCE using an estimated 2.3 million stone blocks, demonstrating sophisticated logistics, geometry, and workforce organization. Roman engineers advanced the field dramatically through the use of pozzolanic concrete — a mixture of volcanic ash, lime, and seawater — enabling the construction of the Pantheon dome (43.3 m diameter, completed around 125 CE) and a vast network of aqueducts and roads across the empire. Cast iron emerged as a structural material during the Industrial Revolution, first used prominently in the Iron Bridge at Coalbrookdale, England, completed in 1779. Wrought iron and later steel allowed far greater spans and heights. The Eiffel Tower, completed in 1889, demonstrated the structural possibilities of wrought iron at scale and influenced the development of steel-frame skyscraper construction in Chicago and New York. Reinforced concrete was systematically developed by Joseph Monier, a French gardener, who patented iron-reinforced concrete pots and panels in the 1860s, and later by engineers including François Hennebique who created the first comprehensive reinforced concrete framing system in the 1890s. The 1906 San Francisco earthquake caused widespread devastation and galvanized the engineering profession to develop seismic design provisions. Subsequent earthquakes — including the 1971 San Fernando and 1994 Northridge events — drove successive improvements in seismic codes, base isolation technology, and ductile detailing of reinforced concrete and steel frames. Building codes became increasingly standardized in the twentieth century, with the International Building Code (IBC) first published in 2000 providing a unified model code adopted across much of the United States. Building Information Modeling (BIM) emerged in the 2000s as a digital workflow integrating architectural, structural, and MEP design into a unified three-dimensional model, fundamentally changing coordination practices across the industry.
Frequently Asked Questions
Formula
tau_max = Tc/J | theta = TL/(GJ) | J = pi*R^4/2 (solid)
Maximum shear stress equals the applied torque times the outer radius divided by the polar moment of inertia. The angle of twist equals the torque times the length divided by the product of shear modulus and polar moment of inertia. For solid circular sections, J equals pi times the radius to the fourth power divided by 2.
Worked Examples
Example 1: Solid Steel Shaft
Problem: Find the maximum shear stress and angle of twist for a solid 100mm diameter steel shaft, 2m long, under 5000 N-m torque. G = 80 GPa.
Solution: J = pi * 50^4 / 2 = 9,817,477 mm4\ntau_max = T*c/J = 5,000,000 * 50 / 9,817,477 = 25.46 MPa\ntheta = TL/(GJ) = 5,000,000*2000/(80,000*9,817,477) = 0.0127 rad = 0.73 deg
Result: tau_max = 25.46 MPa, angle of twist = 0.73 degrees
Example 2: Hollow Shaft Comparison
Problem: Same conditions but with a hollow shaft (100mm outer, 60mm inner diameter).
Solution: J = pi*(50^4 - 30^4)/2 = 8,545,132 mm4\ntau_max = 5,000,000 * 50 / 8,545,132 = 29.27 MPa\ntheta = 5,000,000*2000/(80,000*8,545,132) = 0.0146 rad = 0.84 deg
Result: tau_max = 29.27 MPa, twist = 0.84 deg (13% more stress, 36% less weight)
Frequently Asked Questions
What is torsion in structural and mechanical engineering?
Torsion is the twisting of a structural member when it is loaded by torques (moments) that produce rotation about the longitudinal axis. The resulting shear stresses vary linearly from zero at the center to a maximum at the outer surface for circular sections. Torsion is common in shafts, beams loaded eccentrically, and spandrel beams in concrete frames where floor loads apply twisting moments.
What is the torsion section modulus Zp?
The torsion section modulus Zp (also called the polar section modulus) equals the polar moment of inertia J divided by the outer radius c. It relates the applied torque directly to the maximum shear stress through the formula tau_max = T/Zp. This is analogous to the flexural section modulus S that relates bending moment to bending stress. A larger Zp means a lower maximum shear stress for a given torque.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.
Can I use the results for professional or academic purposes?
You may use the results for reference and educational purposes. For professional reports, academic papers, or critical decisions, we recommend verifying outputs against peer-reviewed sources or consulting a qualified expert in the relevant field.
How accurate are the results from Torsion Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Does Torsion Calculator work offline?
Once the page is loaded, the calculation logic runs entirely in your browser. If you have already opened the page, most calculators will continue to work even if your internet connection is lost, since no server requests are needed for computation.
References
Reviewed by Abdullah, Technical Content Specialist · Editorial policy