Hi,

In this article, we are going to learn how to design a shaft and select a standart bearings and you will also find information about key design. Designing a shaft is a little bit confusing but if you follow the treatments step by step, you can design your shaft for any application.

Before start to design, we must have a purpose. Ask this question yourself;

What will this shaft be used for?

There are different answers for this question according to your application. For instance answers can be like that;

- This shaft will carry a machine element (gears,pulleys etc.).
- This shaft will be used for transmitting power from one point to another(Drive Shaft)

I mean that, before dive into the design, we must examine the all details of our problem. In this example we are going to design a shaft to carry a spur gear.

**Gear Type:**Spur Gear**Module(m):**2mm**Teeth(t):**30**Pressure Angle(alpha):**20 Deg.**Width Of The Gear(b):**55mm**Material:**42CrMo4**Speed(n):**800rpm**Power(P):**5KW*(This power will be transmitted to another gear via this)*

These are the initial parameters for our spur gear. We will suppose our gear is safe under these conditions(5KW power will be transmitted in the speed of 800rpm). As you know Spur gears create radial and tangential forces when they are working synchronously.

At Figure 1, Fr represents the Radial Force, Ft represents the Tangential Force and finally Fn represents the Teeth Force to calcute the value of these forces we use the equations are listed in below.

- Tangential Force Ft=2*Mt/d1
- Radial Force Fr=Ft*tan(alpha)
- Teeth Force Fn=Ft/cos(alpha)

Mt represents the Torsional Moment and d1 is the base diameter of the gear. You can easily calculate both of them.

- Mt=9550*P/n
- d1=m.t

Let’s calculate the values of Fn,Ft and Fr;

- Mt=(9550*5)/800
- Mt=59.68Nm =>rounded to 60 Nm
- d1=2*30
- d1=60mm
- Ft=(2*60*1000)/60
**Ft=2000N**- Fr=2000*tan(20)
**Fr=727.94N =>rounded to 728N**- Fn=2000/cos(20)
**Fn=2128N**

Now we have enough information about the forces will be applied to the shaft. Now we are going to make some predictions.

- Shaft Material AISI1050 CD.
- Shaft will be grinded.
- The power will be transmitted with a key.
- Bearing life(Lh) will be 4000<Lh<12000 hours.
- Bearing will only be exposed to radial loads.
- Width of the bearings(wb) will be grader than 10mm and smaller than 25mm(10<wb<25mm)

The range of the last prediction depends on your experience. If you have no experiance you can use the interval above.

Paper sketches are really important to solve an engineering problem. It provides information geometric requirements.

We define some dimensions on this sketch. These dimensions can change according to your needs but try to keep short the distance between bearings. The main purpose is finding the values of d1,d2,d3,d4 and d5. How do we do that?

Let’s draw a freebody diagram according to this sketch.

Now we have a free body diagram. We know all dimensions and forces. Let’s draw the shear and moment diagram.

As you have seen at the middle of the gear, the bending moment is maximum. The value of maximum bending moment is 410235Nmm. This section also is exposed to torsional moment of 60Nm which is equal to 60000Nmm. This section is the most critical point of the shaft because it is exposed to torsional and bending moment and also have keyseat.

*As you know keyseats, grooves, holes and steps on the shaft increase the stress concentration. *

We are goint to use equations listed in below to calculate the shaft diameter.

As you have seen to calculate the shaft diameter we use long equations and factors are selected from referance tables. Cb which is also known as Ka in shigleys book. This factor is a factor which is selected according to diameter but we are already trying to find diameter 🙂 The equation that we use to find shaft diameter is function of diameter.

You can solve diameter equaition with iterative methods with an initial diameter after a few steps you get the diameter or you can make a prediction about the diameter in the section that you wanna calculate.

To make a good prediction you can use the equation below.

**d=160*(p/n)^(1/3)**

In this equation p represents Power(KW) and n represents speed(rpm). This equation is generated from the equation that we use to calculate the diameter of the shaft is only exposed to torsional moment. There is also awailable calculator in my blog for this equation.

Required minumum diameter for this section is 29.47mm for the shaft is only exposed to torsional moment. In our application there is bending moment too. So lets round the diameter to higher points. Maybe 33mm is enough tho handle with this combined loading, Let’ check;

Our Shaft Material is AISI 1050 CD(Cold Drawn). Properties of this material is listed in below(from Shigley’s Book).

- Re(Yield Strength) : 530MPa
- Rm(Ultimate Tensile Strength) : 630MPa
- Rf=315MPa(0.5*Rm)
- Cb from Table 1 according to diameter is 0,87
- Cy from Table 2 according to Rm and Surface Operation 0,95(Shaft surface will be grinded)
- EK=1.5 (Safety Factor)
- Mb=Mt=60000Nmm(Torsinal Moment)
- Me=41025Nmm (From Shear and Moment Diagram)
- Rfd=?

Rfd will be calcuated according to equation in below;

Rfd=Cb*Cy*Rf/Kç

What is Kç? Kç is also known as Kf and can be calculated from this equation

Kç=Kf=1+q*(Kt-1)

In this equation q is notch sensivity and it is selected from the graph below according to notch radius and Ultimate Tensise Strength(Rm).

We have no information about the keyseat notch radius because we have not yet calculated required key size to transmit power. So we make an assumption like that Kf=Kt.

You can select Kt values from table below.** But this values is suitable for preliminary design calculations. When the shaft geometry is totaly defined. You will make all these calculations according to real q,Kt values for each critical section.**

- Kt=3(End-mill keyseat/Torsional)
- Kç=Kf=Kt
- Rfd=Cb*Cy*Rf/Kç
- Rfd=(0,87*0,95*315)/3
- Rfd=86.7MPa

All parameters are calculated to put them into diameter equation. When we put the values to the equation in below

The result will be 19,5mm. If the result is smaller than the initial diameter(33mm), this section is safe when Safety factor is 1,5.

As you have seen, these calculations take a lot of time. Engineers must know how to handle with this kind of long calculations. I prepared a excel spreadsheet for you. You can download this calc-tool from the link below.

There is also online version of this calculator. Online version is a little bit more powerful. It select Cb and Cy factors automatically but in this article I am going to use excel version for the other critical points.

Before dimensioning the d1,d3,d4 and d5. We must calculate the required key size for d2. We are gonna use DIN6885 A-Type Parallel Key in this aplication.

Keys are the standart machine elements are manufactured according to diameter. Keys are easily selected from reference table with a few calculations. The diameter of the section which the keyseat will be settled on is 33mm. From the referance table below we will select b,h,t1 and t2 values of the key.

**Key Dimensions**

- b=10mm
- h=t1+t2=8mm
- t1=5mm
- t2=3.3mm

H is equal to t1+t2 but the value of t1+t2 is equal to 8,3mm. In reference table h is 8mm. Difference in size of 0,3mm is related with tolerance. It will not effect the calculations that we are gonna do to define the length of the key.

The formula that we use to calculate the L of the key is quite simple P=F/A. We just apply this formula for key,keyway and keyseat.

I have not yet developed a calc-tool for keys. So I am gonna make some hand calculations.

**In this hand calculations for keyway and keyseat T represents surface strength.**

Selecting a bearing is the most easiest part of the design process if you are selecting a bearing which only is exposed to radial loading. Reaction forces at A and B are equal to 1035N and 1095N. I am going to use same bearing for A and B and lets round the radial force to 1100N for both of them.

We select the bearing from the bearings catalogue according to C coefficent which also know as dynamic load rating. There are different types of bearings in the market. In this application I am going to use Deep Groove Ball Bearing.

As I said before, to select a standart bearing we need to calculate dynamic load rating(C). To calculate the required C coefficent you can use the equation below.

L10h which is also known as a bearing life is predicted as 4000<Lh<12000 at preliminary design stage. We will assume L10h value as 7000h. You can select a value for this coefficent according to your maintainance period. I suggest you to select this coefficent from the table below.

I am not gonna do this calculation by hand because I have a online calc-tool for this equation. You can reach this calculator from the link below.

The results will be like that;

Our dynamic load rating value is equal to 7647N under these conditions. Lets select a standart bearing from schaeffler catalogue.

http://medias.schaeffler.de/medias/la/start.do?property&lang=en&mediasS=bps7M8PvEuh_&mediasCall

All these bearings are listed in this page are available for your aplication. You can chose anyone of them. I’m gonna use 6205. Because the inner diameter of this bearing is 25mm. I suppose this value is enough for strength calculation at bearing section. Because at the most critical point, the diameter of the shaft is 33mm. At the same time this bearing is very popular and easily findable in the market and it is cheap 🙂

The product data for 6205 is like at the figure below.

We are going to use same bearing at point A and B. Lets make the shaft geometrically defined according to this inputs.

I added a new keyseat at the end of the shaft. Now we will check the safety of each critical points using excel calctool.

For the section B and F you can assume Kt=3 or you can use the graph below.

**Section A**

- D=33
- d=25
- r=1
- LOADING TYPE – TENSION

Kt:2.2853702868909482

—————–

—————–

**LOADING TYPE – BENDING**

**Kt:2.223786960514233**

—————–

—————–

LOADING TYPE – TORSION

Kt:1.7230420736288508

—————–

—————– - q=0.7(from notch sensivity graph)

Kt was calculated with this tool. We will use the Kt for bending in excel calctool.

Stress Concentration Factor Calculator For Circular Stepped Shaft

I did not do these calculations for each section because the geometric properties sections are similar. I will only calculate the B,C and F

Finally we must calculate the deflection and critical speed of the shaft but I will focus on these calcs in another article.

I hope it is helpful for you 🙂

Have a science day.

Gokberk OZCICEK

*Referances*

*Shigley’s Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering) 10th Edition**Makina Elemanlarının Projelendirilmesi BOZACI A.,COLAK O., KOCAS İ.*