A shaft is the component of a mechanical device that transmits linear or rotational
motion and power. It is integral to any mechanical system in which power is transmitted
from a prime mover, such as an electric machine or an engine, to other parts of the
system. The shaft of transmission equipment has suffered from the fatigue such as
axial, bending and torsion stress depends on its application field. The fatigue in
linear or rotating shafts is a phenomenon that has been known and studied for nearly
a century[1]. Thus, it is necessary to analyze fatigue performance for long life cycles of utilization.
3.2 Fatigue Failure
Failure from fatigue is statistical in nature as much as the fatigue life of a particular
specimen cannot be precisely predicted but rather the likelihood of failure based
on a large population of specimens[2]. This fatigue is caused by repeated cycling of the loads. The fatigue strength is
dependent on the type of loading (axial, bending, or torsion) and it has resulted
in three separate fatigue strengths being defined. In particular, this linear motion
is implemented by designing shafts using axial and bending stress among the three
fatigue strengths.
The state of stress to be considered is caused by force (torque) transmitted to the
shaft, axial forces imparted to the shaft, bending of the shaft due to its weight
or loads, and torsion by rotating of shaft. The characteristics of these three basic
types or cases are given here : axial strength $\left(\sigma_{a}\right)$, bending
strength $\left(\sigma_{b}\right)$, and torsion strength $\left(\sigma_{t}\right)$.
그림 1. 축 방향 응력, 굽힘 응력 및 비틀림 응력
Fig. 1. Three Strengths ; Axial, Bending, and Torsion
Axial Strength
Axial stress is referred to as normal stress since it acts in line with the material.
Most simple form is tension stress since its effect is independent of the length of
the material. The formula for definition of axial stress can be expressed as follows.
where, $A_{rea}$ is cross-section area of the shaft, and $F_{\max}$ represents the
maximum force considering mechanical load given by equation (9).
Bending Strength
The first step in shaft design is to draw the bending moment diagram for the loaded
shaft or the combined bending moment diagram. The bending stress is larger than the
direct stress by axis when a shaft is easily subjected to bending moment in such a
linear motion case. From the bending moment diagram, the points of critical bending
stress can be determined.
The force and moment acting on shaft indicates equation (13) and (14), respectively.
where, $g$ is gravity acceleration.
The effective nominal stress is [3].
Torsion Strength
Torsion occurs when any shaft is subjected to a torque in case of a rotating motion
in general. This is true whether the shaft is rotating (such as drive shafts on engines,
motor and generators) or stationery (such as with a bolt or screw). This torque makes
the shaft twist and one end rotates relative to the other inducing shear stress on
any cross-section.
It represents as the equation (16) for circular solid[4].
where, $T_{m}$ is close to zero, which represents the twisting moment. In this study,
the torsional strength is excluded because it has little impact on the shaft due to
the linear motion.
Total Strength
The total strength acting on the shaft is expressed by sum of the axial and bending
strength.
Woehler Curve
First systematic study to characterize the fatigue behavior of materials cyclic stress
range was conducted by Woehler[5]. In high-cycle fatigue situations, materials performance is commonly characterized
by an S-N Curve, also known as a Woehler curve. This is a graph of the magnitude of
a cyclic stress against the logarithmic scale of cycles to failure.
Alternating Strength by Woehler Curve
By Woehler curve, we can find the endurance limit value of selected material; it is
marked as $\sigma_{en}$ in failure cycle $10^{6}< N < 10^{7}$.
The safety factor, $\zeta$, is a ratio of maximum strength to intended loads for the
actual item that was designed. The value to use for safety factor is based on judgment.
It depends on the consequences of failure, that is, cost, time, safety, etc. Some
factors to consider when selecting a value for safety factor are how well the actual
loads, operating environment, and material strength properties are known, as well
as possible inaccuracies of the calculation method. Values typically range from 1.3
to 6 depending on the confidence in the prediction technique and the criticality of
the application. Unless experience or special circumstances dictate it, the use of
safety factor values of less than 1.5 is not normally recommended[3].
Summary
Its features for this study may be summarized as follows ;
The Woehler curve shows fatigue life corresponding to a certain stress amplitude
The Woehler diagram can be used to design for finite (and infinite) life
For steel, the fatigue limit corresponds to $10^{6}< N < 10^{7}$