The 45 degree rule" and its impact on strength and stiffness of a shaft subjected to a torsional load "

Date of Award


Degree Name

M.S. in Mechanical Engineering


Department of Mechanical and Aerospace Engineering


Advisor: Thomas James Whitney


Many industrial machines incorporate a multitude of moving and rotating parts necessary for the machinery to perform its intended functions. Rotating machinery, like turbines and compressors, include multiple parts that rotate under heavy loads and high speeds. Shafts are a common medium to transmit these loads and speeds. Quite often, these shafts are required to be stepped to create multiple distinct diameters for carry and located other components. The addition of these steps must be design with care such that a proper radius is selected between two diameters. Parts operating in this field will run for long periods of time and must maintain under multiple start/stop cases and can eventually cause failures. Rotor and torsional dynamic analyses are completed on most if not all rotors in the turbomachinery field. The 45 degree rule is a method of simplification for modeling abrupt changes in diameter. This rule of thumb states a line from the lesser of two steps on a shaft can be drawn at a 45 degree angle to the outside diameter of the greater step. The material outside this line can be modeled with zero modulus and actual density. This region of material does not significantly impact the torsional stiffness of the area. The purpose of this research is to find the effect this modeling approach has on the computed strength, stiffness, and overall rotordynamic properties of a rotating shaft. This will also demonstrate the Best case" shoulder combination with various fillet radii and/or other angle orientation as well as illustrate additional areas this theory may be applicable. Additional considerations will be made for defective or non-homogeneous material (e.g., inclusions, cracks, and scratches) that may be contained within the region under consideration and their effect on the overall region's computed strength and stiffness. The purpose of this research is split into three claims. The first claim states that after the removal of the material outside the 45° line, neither the strength nor stiffness will be significantly impacted analyzed at various fillet sizes. This claim was proven to be true. After comparing the stress concentration factors for both 90° and 45° geometries, there is no significant (worsening) impact to the stress concentration factor. The stress concentration factor for the 45° design on average reduces the stress concentration factor. The second claim is to prove the assumption of the 45° rule that the stiffness grows or shrinks along the 45° line. This is proven to be false due to the stiffness significantly increasing as the taper of the should progresses from 0° (90°) step to 60°. The application of the rule will deem whether this difference is significant. The third claim states that if the first claim is proven true, the impact of non-conformities, holes and cuts, inserted into the 45° region will not significantly impact the stress concentration factor. This is proven true, it was seen there is not a significant impact to the stress concentration factor when adding holes or cuts into the 45° region. It can be noted that if the 45° boundary is passed, there will be an increase in stress in the fillet at the hole location."


Shafting Design and construction, Torsion, Strains and stresses, Strength of materials, Mechanical Engineering, Rotordynamic, 45 degree rule, torsional analysis, tapered shaft, rotating machinery, shaft shoulder design, turbo-machinery, strength, stiffness, FEA, finite element analysis, mesh sensitivity, deflection, stress concentration factor, rotor balancing

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