# Angular deflection of hollow shaft

Calculate the angular deflection of a hollow shaft with this free calculator. Quickly and easily find the angular deflection of your shaft in a few simple steps.

When an expansion joint bends around its center, located on the centerline and halfway between the ends of the bellows, it is known as angular deflection. To calculate the angular deflection of a hollow shaft, you can use the online Angular Deflection of Hollow Cylinder Calculator. All you need is the Applied Torque (T), Shear Modulus (G), Outside Diameter (D), Inside Diameter (d), and Unsupported Length (L).

Applied Torque
N-mm
Shear Modulus
MPa
Outside Diameter
mm
Inside Diameter
mm
Unsupported Length
mm
Send the result to an email

2 Number of calculations
\alpha = \frac{32 LT}{G\pi (D_4 - d_4)}

where:

• α is the deflection of the shaft
• T is the applied torque
• L is the unsupported length of the shaft
• G is the shear modulus of the shaft material
• D is the outside diameter of the shaft
• d is the inside diameter of the shaft.
The formula for the deflection of a shaft under torsion takes into account several factors that affect the amount of deflection that will occur in a shaft when it is subjected to a twisting force or torque.
• T: The applied torque is the twisting force that is applied to the shaft. It is measured in units of force multiplied by distance, such as newton-meters or pound-feet.
• L: The unsupported length of the shaft is the distance between the points where the shaft is supported. If the shaft is supported at both ends, then the unsupported length is equal to the total length of the shaft.
• G: The shear modulus of the shaft material is a measure of the material’s resistance to shearing or twisting forces. It is measured in units of pressure, such as pascals or pounds per square inch.
• D and d: The outside diameter and inside diameter of the shaft respectively are the dimensions of the shaft’s cross-sectional area. The difference between the two is known as the wall thickness of the shaft.
The formula tells us that the deflection of the shaft is directly proportional to the applied torque and the unsupported length of the shaft, and inversely proportional to the shear modulus of the shaft material and the fourth power of the difference between the outside and inside diameters of the shaft. This means that a larger torque or unsupported length will result in a larger deflection, while a higher shear modulus or thicker wall will result in a smaller deflection.