## What is a shaft?

The shaft is basically defined as the machine element that rotates, is in circular cross-section, it is majorly used to transfer the power from one part of the system to another, or from a power generating machine to a power absorbing machine. The shaft is supported on bearings and is rotated with the help of a set of gears or pulleys that are used for the power transmission. Bending moment, torsion, and an axial force act upon the shaft. Designing of shaft primarily involves determining stresses at a critical point in the shaft that is arising due to loading.

## Types of shafts

The following two types of shafts are important from the subject point of view:

### Transmission shafts

The transmission shafts are the shafts that are responsible for the transmission of the power between the source and the power absorbing machine. The countershafts, line shafts, overhead shafts, and all factory shafts are transmission shafts. In addition to twisting, these shafts are subjected to bending as they carry some other machine parts like the pulley and gears.

### Machine Shafts

These shafts form an integral part of the machine itself. The crankshaft is an example of a machine shaft.

## Shaft materials

Various materials like ferrous materials, non-ferrous materials, and non-metals are all used for manufacturing the shafts depending on the use of the shaft. Some of the metals that are used for the manufacturing of the shafts are discussed as follows:

### Hot-rolled (HR) plain carbon steel

Hot-rolled plain carbon steel is one of the cheapest materials that is used for the manufacturing of shafts. Since the material is hot rolled, the scaling is always present on the surface of the materials, and some machining is needed in order to make the surface of the material smooth.

### Cold-drawn (CD) plain carbon or alloy composition

The material that is cold-drawn basically has inherent characteristics like the surface of such materials are smooth and have a bright finish. The machining process that is utilized for making the surface smooth is minimum. In these materials, you are able to see better yield strength, and these types of materials are widely used for general purpose transmission shafts.

### Alloy steels

Alloy steel is a mixture of various elements with the parent steel to improve certain physical properties. To retain the total advantage of alloying materials, one requires heat treatment of the machine components after they have been manufactured. Nickel, chromium, and vanadium are some of the common alloying materials. However, alloy steel is expensive.

These types of materials are generally used in conditions that are relatively harsh in service. These shafts are used where the situation demands more strength. There are very few chances of these shafts getting cracked or distort during the heat treatment. In these shafts, the residual stress is also less as compared to the shafts made up of carbon steel.

In certain cases, the shaft needs to be wear-resistant, and then more attention has to be paid to make the surface of the shaft wear-resistant. The common types of surface hardening methods are,

- Hardening of surface
- Case hardening and carburizing
- Cyaniding and nitriding

## Some important points regarding shaft material

- Deflection in the shaft is primarily controlled by geometry, not material.
- Stress in the shaft is controlled by geometry, not material.
- Strength of a shaft is controlled by its material property.
- Shafts are commonly made from low carbon, CD, or HR steel, such as AISI 1020-1050 steel.
- Fatigue properties donâ€™t usually benefit much from high alloy content and heat treatment.
- Surface hardening is usually used when the shaft is being used as a bearing surface.
- Cold drawn steel typical for, d < 3 inches.
- HR steel is common for larger size shafts and should be machined all over.
- For low production quantities, lathe machining is typical. Therefore, minimum material removal may be the design goal.
- For high production quantities, forming or casting is common. Therefore, minimum material may be the design goal.

## Stresses in shafts

The following stresses are induced in the shafts:

- Shear stresses due to the transmission of torque (i.e., due to torsional load)
- Bending stresses (tensile or compressive) due to the forces acting upon machine elements like gears, pulleys, etc. as well as due to the weight of the shaft itself.
- Stresses due to combined torsional and bending loads.

## Design considerations for shaft

For the design of the shaft following two methods are adopted,

### Design based on strength

In this method, design is carried out so that stress at any location of the shaft should not exceed the material yield stress. However, no consideration for shaft deflection and shaft twist is included. In designing shafts on the basis of strength, the following cases may be considered:

### Shafts subjected to bending moment only,

$M=\frac{\mathrm{\pi}}{32}\times {\sigma}_{b}{\left({d}_{0}\right)}^{3}\left(1-{k}^{4}\right)$

where $M$ is the bending moment at the point of interest, ${d}_{0}$ is the outer diameter of the shaft, $k$ is the ratio of inner to outer diameters of the shaft ($k=0$ for a solid shaft because inner diameter is zero).

### Shafts subjected to axial stress only,

${\sigma}_{a}=\frac{4\alpha F}{{\mathrm{\pi d}}_{0}^{2}\left(1-{\mathrm{k}}^{2}\right)}$

where, $F$ is the axial force (tensile or compressive) and $\alpha $ is the column-action factor.

### Shafts subjected to combined bending and axial stress,

${\sigma}_{x}=\left[\frac{32M}{{\mathrm{\pi d}}_{0}^{3}\left(1-{\mathrm{k}}^{4}\right)}+\frac{4\alpha F}{{\mathrm{\pi d}}_{0}^{2}\left(1-{\mathrm{k}}^{2}\right)}\right]$

### Shafts subjected to axial loads in addition to combined torsional and bending loads,

Tensile stress due to axial load is given by,

${\sigma}_{t}=\frac{P}{A}$

where $P$ is axial load acting on the shaft,Â $A$ is the cross-sectional area of the shaft.

As the nature of the bending stress and axial stress is same, these can be vectorially added for any location on the shaft, to get the resultant tensile or compressive stress, which can be used to find the principal stresses in the shaft.

## Design based on stiffness or rigidity

The basic idea of design in such a case depends on the allowable deflection and twist of the shaft.

### Torsional rigidity

For a shaft subjected to twisting moment, the angle of twist is given by,

$\theta =\frac{TL}{GJ}\le \left[\theta \right]$

where *T* is torque applied,Â *L* is the length of the shaft, *J* is the polar moment of inertia of the shaft about the axis of rotation and *G* is the modulus of rigidity of the shaft material.

Therefore, for the known values of *T, L,* and *G* and allowable value of angle of twist, the diameter of the shaft can be calculated.

### Lateral rigidity

Bending moment acting on any shaft is given by,

$M=EI\frac{{d}^{2}y}{d{x}^{2}}$

By integrating this equation twice with respect to $x$ and by applying the boundary conditions, $y$ can be calculated.

## Critical speed of shafts

The theoretical angular velocity that is responsible to excite the natural frequency of a rotating object like a shaft is known as critical speed. When the speed of the shaft approaches the natural frequency of the object, resonance in that object starts, which leads to an increase in the vibrations of the system. The resulting resonance occurs regardless of orientation.

Critical speed can also be defined as when the rotational speed is equal to the numerical value of the natural vibrations. The lowest rotational speed at which this natural vibration occurs is called the first critical speed.

For any small deflection for a rotating shaft, there is a speed at which the centrifugal force is equal to the elastic restoring force, at this point of time the deflection in the shaft increases greatly and the shaft is said to whirl. This effect is very much reduced if there is a change in this speed. This critical speed (whirling speed) is dependent on the shaft dimensions, the shaft material, and the shaft loads.

## Common Mistakes

The other two similar forms of a shaft are axle and spindle. The axle is a non-rotating member used for supporting rotating wheels etc. and does not transmit any torque. A spindle is simply defined as a short shaft. However, the design method remains the same for axle and spindle as that for a shaft.

## Context and Applications

This topic is significant in the professional exams for both graduate and postgraduate courses, especially for

- Bachelor of Technology in Mechanical Engineering
- Master of Technology in Machine Design
- Master of Technology in Mechanical Engineering

## Related Concepts

- Von Mises stresses
- Soderberg equation
- Flywheel
- Slope and deflection in a beam
- Shear stress and bending stress in a beam

## Practice Problems

**Q1**. Is ASME Standard uses shock and fatigue factors in the machine design consideration?

- Yes
- No

Correct Option: **(a)**

**Explanation:** The moment is specifically multiplied by a specific number to consider the shock and fatigue factors for designing any shaft. The moment types are bending and twisting moments multiplied by those factors for the evaluation of dynamic loading, equivalent bending, and twisting moments.

**Q2**. The stiffness of a solid shaft is considered more than the stiffness of a hollow type of shaft with the same weight.

- True
- False

Correct Option:** (b)**

**Explanation:** The center material will be removed from the center in the hollow shaft. The material is evenly distributed at a large radius. The hollow shaft is extra stiffer than the solid shaft.

**Q3**. Flexible shafts are flexible as they have _______ rigidity in torsion.

- Low
- High
- Very high
- Infinitely small

Correct Option: **(b)**

**Explanation:** Flexible shafts are highly rigid in torsion. Due to this, the shafts are capable of transmitting power and torque. Flexible shafts are used to transfer the torque via bends and circular curves.

**Q4**. Flexible shafts are considered to have _______ amount of rigidity in the bending moment.

- High
- Low
- Very high
- Extremely low

Correct Option: **(b)**

**Explanation:** Flexible shafts are less rigid in the bending moments, and thus it helps them become more flexible. The flexible shafts are used to transmit the rotary positions between two objects.

**Q5**. From the ASME design criteria code, the maximum allowable shear stress is considered as X% of yield strength or Y% of ultimate strength.

- X=30 Y=18
- X=30 Y=30
- X=18 Y=18
- X=18 Y=30

Correct Option: **(a)**

**Explanation:** Allowable stress is defined as the maximum allowable stress in designing the system and working stress analysis. For ASME Standard, the lower value is considered in the given two values.

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