Angular Deflection of Hollow Shaft Calculator

Welcome to this engineering tutorial where we will explore the fascinating concept of angular deflection of a hollow shaft and its calculator. Angular deflection is an essential parameter in various mechanical and structural applications. We will discuss the formula, provide an example of its real-life usage, and delve into some interesting facts about angular deflection.

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Select Unit
Applied Torque (T) N-mm
Shear Modulus (G) Mpa
Outside Diameter (D) mm
Inside Diameter (d) mm
Unsupported Length (L) mm
Angular Deflection of Hollow Shaft Calculator Results
Deflection of shaft = radians

1. Understanding Angular Deflection

Angular deflection refers to the amount of rotation experienced by a shaft or structural member when subjected to an external moment or torque. It measures the angular displacement or twist in degrees or radians that occurs along the length of the shaft. Angular deflection is a crucial consideration in designing shafts, especially in cases where excessive twisting can lead to failure or undesirable performance.

2. The Formula

The angular deflection of a hollow shaft can be calculated using the following formula:

θ = (T * L) / (G * J)


  • θ is the angular deflection in radians
  • T is the applied torque or moment in Nm (Newton-meters)
  • L is the length of the shaft in meters
  • G is the shear modulus of the material in Pa (Pascals)
  • J is the polar moment of inertia of the hollow shaft in m4

3. Real-Life Application

To illustrate the practical application of the angular deflection of a hollow shaft, let's consider the construction of a wind turbine. Wind turbines convert the kinetic energy of wind into mechanical energy, which is then used to generate electricity. The rotor of a wind turbine is connected to the main shaft, which transfers the rotational energy to the generator.

By using the angular deflection formula, engineers can determine the appropriate dimensions and materials for the turbine shaft. The formula helps them calculate the expected amount of twist and select a hollow shaft design that can withstand the torque generated by the wind turbine rotor. This ensures the shaft remains within acceptable deflection limits, preventing excessive stress and potential failure.

For example, suppose we have a wind turbine with an applied torque of 1000 Nm, a shaft length of 10 meters, a shear modulus of 80 GPa, and a polar moment of inertia of 2.5 × 10-4 m4. Using the formula, we can calculate the angular deflection as follows:

θ = (1000 * 10) / (80 * 2.5 × 10-4)

After performing the calculation, we find that the angular deflection of the wind turbine shaft is approximately 0.5 radians. This information helps engineers ensure that the selected hollow shaft can handle the expected twisting without compromising the turbine's performance and structural integrity.

4. Interesting Facts

Now let's explore some interesting facts about angular deflection and its significance in engineering:

  1. Torsional Rigidity: Angular deflection is closely related to the torsional rigidity of a shaft. Torsional rigidity measures a shaft's resistance to twisting and is influenced by factors such as material properties, cross-sectional geometry, and length. By understanding and controlling angular deflection, engineers can ensure the desired level of torsional rigidity in mechanical systems.
  2. Structural Stability: Excessive angular deflection can lead to structural instability in various applications. For example, in high-rise buildings or bridges, shafts and beams must maintain minimal deflection to prevent the structure from swaying or vibrating excessively under external forces such as wind or earthquakes. Precise calculations of angular deflection aid in designing stable and safe structures.
  3. Shaft Material Selection: The choice of material for a hollow shaft significantly impacts its angular deflection. Different materials have varying shear moduli (G) and affect the shaft's overall rigidity. Engineers carefully consider material properties, such as strength, stiffness, and ductility, to ensure the shaft can withstand the applied torque while keeping deflection within acceptable limits.
  4. Dynamic Applications: Angular deflection is particularly relevant in dynamic systems where continuous or fluctuating torque is present. Examples include rotating machinery like engines, turbines, and motors. Understanding the angular deflection characteristics allows engineers to optimize performance, minimize energy loss, and prevent premature wear and failure of critical components.
  5. Composite Materials: The use of composite materials in hollow shafts is gaining popularity due to their high strength-to-weight ratio and tailored stiffness properties. However, calculating angular deflection in composite shafts is more complex compared to traditional materials. Advanced analytical and numerical techniques, such as finite element analysis, are employed to accurately predict deflection behavior and ensure optimal design.

By considering these interesting facts and applying the angular deflection formula, engineers can make informed decisions, prevent structural failures, and design efficient and reliable mechanical systems.


In this tutorial, we explored the concept of angular deflection of a hollow shaft and its calculator. We discussed the importance of angular deflection in various engineering applications, provided the formula for calculating it, and gave an example of its real-life usage in wind turbine design. Additionally, we learned interesting facts about angular deflection and its impact on structural stability, material selection, and dynamic systems.

Understanding and accurately calculating angular deflection allows engineers to ensure the safe and efficient operation of mechanical systems. By considering factors such as torsional rigidity, material properties, and dynamic loads, they can design shafts that minimize deflection and prevent undesirable effects such as excessive stress, vibration, and failure.

Remember, the angular deflection of a hollow shaft is just one of many crucial considerations in engineering design, and it plays a significant role in ensuring the reliability and performance of mechanical systems.