Stepped Shaft Automatic Design Considering a New...

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 155

202801-5757-IJMME-IJENS © February 2020 IJENS I J E N S

Stepped Shaft Automatic Design Considering a

New Integrated Technique of Stress, Deflection, and

Critical Speed Analysis

Hesham A. Abdou*, Sara A. El-Bahloul, Tawakol A. Enab, and N. Fouda

Production Engineering & Mechanical Design Department, Faculty of Engineering, Mansoura University

El-Gomhoureya Street, Mansoura 35516, Dakahlia, Egypt

* E-mail: [email protected]

Abstract-- Since the shaft is an important part of all

machines, it is deemed very necessary to design it effectively.

The whole shaft design involves several iterative steps

beginning with the determination of the shaft's preliminary

dimensions. This paper introduces software that is based on a

new integrated method of stress, deflection and critical speed

analysis to determine the optimal dimensions of different

stepped shafts. This software was developed with the help of a

graphical user interface and visual aids using MATLAB as a

programming language. In order to determine the optimal

shaft diameter, the fatigue stress analysis is conducted at the

most critical locations automatically. Analysis of deflection is

then performed to check whether the resulting dimensions are

safe or not. The study of vibration is also conducted to test

whether the shaft runs safely away from critical ones.

Verification is also carried out on the basis of a comparison

between the proposed design and the Shigley design on a given

case study, which shows that the new approach used in the

proposed design leads to more accurate results. The proposed

software greatly serves any user who has no experience or

knowledge of different design techniques and strategies,

resulting in time, effort, and cost savings.

Index Term-- Automatic design, stepped shaft, fatigue stress,

deflection, critical speed, and stress concentration.

1. INTRODUCTION

A shaft is a rotating machine part that is typically a circular

cross-section used for power transmission. In order to

transfer the necessary torque, a shaft rests on bearings and

holds other machine parts such as gears, pulleys, flywheels,

sprockets, and cranks. Such machine parts produce loads on

the shaft of bending, axial, and torsion. Additionally, Most shafts are usually stepped and having keyways, sharp

corners, etc., which introduce fatigue cracks that eventually

propagate in some highly stressed regions until a final

fracture occurs. Therefore, the shaft design must maintain

the static and dynamic loads produced in order to achieve

durability [1-3].

In order to obtain the most durable shaft design, extensive

research has been carried out. In the analysis of stepped

shaft, Egelhoff et al., used modern engineering tools to solve

a structured problem using energy techniques [4].

Marudachalam et al., used the Goodman approach to

improve shaft design under fatigue loading [5]. Gujar et al.,

designed a multi cross-section shaft using only the adapted

Goodman principle and compared with Finite Element Analysis (FEA) [3]. Shivakumar et al., made a FEA of lathe

spindle using ANSYS [6]. Ravinder et al., studied the gear

shaft by analyzing the stresses, torques and bending

moments by developing a Computer-Aided Design (CAD)

model in Pro-E software, and conducting a FEA utilizing

ANSYS software [7]. Tavares and de Castro compared the

fatigue analysis methodologies of shafts based on the

Soderberg criterion [8]. Armah used the ASME code

methodology of fatigue strength analysis to construct a

power transmission shaft at a constant speed during fatigue

loading [1]. From the literature review it is apparent that less

discussion and resreaches have been performed on stepped shaft automatic full design based on stress, deflection, and

critical speed analysis together.

The aim of this research is to automatically determine the

optimal design of a stepped shaft based on a new integrated

approach of stress, deflection, and critical speed analysis.

This software is developed with the help of a graphical user

interface and visual aids using MATLAB as a programming

language. Additionally, to aid with effective visualization, a

graph of the optimal shaft layout is accomplished. The

proposed software will greatly serves any user who has no experience or knowledge of different design techniques and

strategies, resulting in time, effort, and cost savings.

2. AN INTEGRATED PROCEDURE FOR STEPPED SHAFT

DESIGN

Figure 1 shows the integrated stepped shaft design

technique that will be used to automatically calculate the

optimal dimensions of stepped shafts, based on new

integrated approach of stress, deflection, and critical speed

analysis.

Firstly, it is important to provide a general labeled

layout for the shaft with dimensions. The second step is to

identify any unknown loads to estimate the reactions at the

supports using equations of static equilibrium.

mailto:[email protected]



Fig. 1. The integrated procedure for the shaft design process

Accordinally, the shear force diagram (SFD), bending

moment diagram (BMD) and torque diagram (TD) are

drawn in two planes and then combined to identify the most critical location. The third step is to identify the shaft

material properties and the factor of safety. If the shaft

material is undetermined, a material of lower strength is

applied first. The fatigue strength (Se) is determined for an

infinite life shaft. For the factor of safety (FS), a minimum

desired one is specified first. Afterward, the goal is to

determine automatically the values of modification factors

used in the calculation for all potential critical locations

where higher stress exists. The values of surface factor (ka),

loading factor (kc), temperature factor (kd), reliability factor

(ke) and miscellaneous factor (kf) will be determined once and there is no need to change them later during iterations.

But the values of Size Factor (kb) and fatigue stress

concentration factor (Kf) will be determined first with

typical values and will be refined later during iterations for

more accurate design. To get preliminary shaft dimensions,

appropriate fatigue design equations are applied for the

most critical point to express diameter size in terms of Kf

and kb. [1]

Safe

Not Safe

Change

material

Load Analysis

Material Properties and Factor of Safety

Stress Analysis

Design Equation

Solution to get preliminary shaft dimensions

Check of other shaft critical points

Deflection Analysis

Vibration Analysis

Shaft Layout

Refine

constant

s

Safe

Not Safe

Change

dimension

s

Final result

Safe

Not Safe



To refine the decision choice, a deflection analysis is

performed. Singularity functions are used to determine

deflection as they are excellent for managing discontinuities.

If the deflection analysis is unsafe, the values of diameters

are changed by multiplying them with the largest

magnification factor resulted, consequently other related

dimensions change through appropriate proportions. By

determining the FS for all critical locations, the design is

safe at a certain location if its FS is larger than the desired

value. If the FS is less than the desired value, then failure

will happen before the desired number of cycles (N).

Therfore, higher strength material can be used in this case.

[2]

The shaft critical speeds are determined by performing

vibration analysis. Critical speeds are associated with

uncontrolled large deflections, which occur when inertial

loading on a slightly deflected shaft exceeds the restorative

ability of the shaft to resist. Shafts must operate well away

from such speeds. Since most shafts are of variable

diameter, Rayleigh’s equation will be more useful for estimating the first critical speed, treating simultaneously

the contributions of concentrated masses (e.g., gears,

pulleys, sprockets, cams, etc.) and the distributed shaft mass

as well. Additionally, Singularity functions method is used

for determining the deflection of the shaft to be used in

critical speed calculations. [2, 9]

3. PROPOSED SOFTWARE INTERFACE

The proposed software automatically determine the optimal

design of a stepped shaft based on a new integrated

approach of stress, deflection, and critical speed analysis.

This software is developed with the help of a graphical user interface and visual aids using MATLAB as a programming

language. The software greatly serves any user who has no

experience or knowledge of different design techniques and

strategies, resulting in time, effort, and cost savings. Figure

2 shows the performed software input interface where the

user can easily enter all needed input values such as axial

dimensions, acting forces, shaft speed, gear and bearing

dimensions and types, gear specific weight, desired safety

factor, temperature, reliability, surface finish, and starting

material properties. Then, the user can easily press the

CALCULATE button to begin the automated shaft design technique according to the integrated procedure for shaft

design previously discussed and illustrated in Fig. 1. It is

obvious that the software interface is designed to facilitate

the interaction of the user with the software.

4. SOFTWARE IMPLEMENTATION

In order to show in details the processing manner of the

software according to the previous procedure, a case study

taken from ‘Shigley's mechanical engineering design’ is

implemented. A shaft is used as a part of a double reduction

gearbox. The general layout and axial dimensions of the

shaft are illustrated in Fig. 3. The shaft is running at a

constant speed of 3000 rpm. It carries two spur gears at

points (G) and (J) and is supported simply by bearings (A) and (B). It is required to determine the appropriate

diameters for each section of the shaft for infinite life of it,

by selecting a suitable material and desired FS equal to 1.5.

[2]

The user can easily enter the required inputs in the software

interface as shown in Fig. 2. A cold drawn (CD) steel

( AISI 1020) material is initially chosen with Ultimate

tensile strength (Sut) = 68 kpsi and the desired FS = 1.5. The

material selected has an elasticity modulus (E) equal to 30

Mpsi.

4.1. Preliminary shaft dimensions calculations

The SFD, BMD, TD, and total BMD are automatically

performed using equations of static equilibrium as ilustrated

in Fig. 4. From the total BMD, it is clear that the critical location without considering the effect of stress

concentrations is found at point (J).

The Goodman method is used at point (I) to get the

preliminary shaft dimensions then making iterations to

refine them. Modification factors are determined

automatically as follows: ka=0.883, kc=1, kd =1, and

ke=0.753. For the constant kb, it is assumed first as 0.9, and

then it is refined later when the diameter is known. At

location (I), the software calculates automatically the

remaining constants as follow: theoretical stress

concentration factor for normal stress (Kt) = 1.5987 and for

shear stress (Kts) = 1.3304 and notch sensitivity for normal stress (q) = 0.7868 and for shear stress (qs) = 0.8314.

Accordinally, the fatigue stress-concentration factor for

normal stress (Kf) = 1.4711 and for shear stress (Kfs) =

1.2747. The first estimation of the small diameter of the

shoulder at point (I) is 1.6096 in. The iterations for the

solutions will include the parameters (kb, q, Kt, qs, Kts, Kf,

Kfs, and Se) to improve the results by refining the

calculations until achieving constant diameter for three

decimal places. After the final iteration (when the diameter

difference is smaller than 0.001) the dimensions at location

(I) will be as follow: the small diameter of the shoulder (D5)

= 1.62 in, fillet radius (r5) = 0.162 in and the large diameter of the shoulder (D4) = 1.94 in. The obtained preliminary

stepped shaft diameters on the basis of fatigue stress

analysis are: D1=1.13 in, D2=1.35 in, D3=1.62 in, D4=1.94

in, D5=1.62 in, D6=1.35 in, and D7=1.13 in.



Fig

. 2

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he p

rop

ose

d s

oft

war

e i

nte

rfac

e



Fig. 3. General shaft layout (Dimensions are in inches)

Fig. 4. Shaft load analysis diagrams

4.2. Deflection analysis

After determining the preliminary shaft dimensions,

deflection analysis is applied to check the deflections and

slopes at gears and bearings location in order to control the

transverse deflection and the angular deflection (slope)

within an acceptable range. Otherwise, changes in the

dimensions should be performed to resolve this problem.

As the shaft hasn’t the same cross-section along its length,

a detailed analysis using the singularity functions method

must be used to give more accurate results compared to

the simplified analysis.

The simplified analysis can be performed by assuming a

uniform cross-section of the shaft with minimum diameter.



The shaft slopes and deflections are obtained by modeling

the shaft twice with loads in two orthogonal planes using

equations (1) and (2) for xy-plane and equation (3) and (4)

for xz-plane. Then the results can be combined through an

orthogonal vector. Details like grooves and keyways are

ignored as they will have an insignificant effect on the deflections. It has been noticed that the stepped shaft will

lead to different second moment of inertia at the transition

sections, so deflection and slope have two unreal results at

the same location. Consequently, the results are not

accurate so one can either simplify the diameter to only one

value (smallest one), acordinlly; the second moment of

inertia is the same. Otherwise, use the singularity functions method to overcome this problem.

dy

dx =

(1

2∗356.725∗⟨x⟩2)−(

1

2∗197∗⟨x−2⟩2)−(

1

2∗885∗⟨x−7.75⟩2)−(4.0963∗103)

30∗106∗I1 (1)

y = (

1

6∗356.725∗⟨x⟩3)−(

1

6∗197∗⟨x−2⟩3)−(

1

6∗885∗⟨x−7.75⟩3)−(4.0963∗103∗x)

30∗106∗I1 (2)

dy

dx =

(1

2∗114.975∗⟨x⟩2)+(

1

2∗540∗⟨x−2⟩2)−(

1

2∗2431∗⟨x−7.75⟩2)−(6.0627∗103)

30∗106∗I1 (3)

y = (

1

6∗114.975∗⟨x⟩3)+(

1

6∗540∗⟨x−2⟩3)−(

1

6∗2431∗⟨x−7.75⟩3)−(6.0627∗103∗x)

30∗106∗I1 (4)

By using the singularity functions method, the bearing

reaction forces can be calculated to get the load intensity

equation, and numerically integrated to generate equations

for shear force, bending moment, slope and deflection

respectively. Figure 5 shows the diagrams of shear force,

bending moment, slope and deflection equations for xy-

plane and similarly for xz-plane. For the xy-plane, as (Ii) is the second moment of inertia for (ith) shaft section is not the

same (where i = 1:7), so the equation of (moment/inertia)

[M/I] can’t be used directly. Hence, some modifications

must be performed firstly. In the current case study, the

stepped shaft under consideration has six steps (w = 1:6)

according to diameter. Each step far of the datum by a

distance lw. For the first step (w =1), the values of (M/I) at

the points (b) and (c), and the step change [Δ(M/I)w=1] are

calculated. Also, the slope (m) for lines (ab) and (cd) are

mab and mcd respectively.The slope change for first step is

[(Δm)w=1] which can be calculated as the slope difference

between mab and mcd. Similarly, [Δ(M/I)w] and [(Δm)w] can

be calculated for the other five steps. Then the modification

is done by dividing the [M/I] equation terms by their (Ii)

and by adding steps [Δ(M/I)w] and ramps [(Δm)w].

Figure 6 shows the plotting of the (moment/inertia) equation

[M/I]. By integrating the equation of (Moment/inertia)

twice with adding previous modification, the detailed

equations of slope and deflection are obtained for xy-plane

using equation (5) and (6) and for xz-plane using equation

(7) and (8) respectively. Accodinly, the slope and

deflection diagrams are plotted for simplified and detailed

analysis methods of both planes as shown in Fig. 7.

dy

dx =

1

30∗106 {

356.725

2∗I1⟨x⟩2−

197

2∗I2⟨x − 2⟩2 −

885

2∗I5⟨x − 7.75⟩2 +

∑ Δ (M

I)

w∗ ⟨x − lw⟩1

w=1:6 + ∑ (1

2∗ Δmw⟨x − lw⟩2

w=1:6 ) − (9.8361 ∗ 103)} (5)

y = 1

30∗106 {

356.725

6∗I1⟨x⟩3−

197

6∗I2⟨x − 2⟩3 −

885

6∗I5⟨x − 7.75⟩3 +

∑ (1

2∗ Δ(M/I)w ∗ ⟨x − lw⟩2

w=1:6 ) + ∑ (1


w=1:6 ) − (9.8361 ∗ 103 ∗ x)} (6)

dy

dx =

1

30∗106 {

114.975

2∗I1⟨x⟩2+

540

2∗I2⟨x − 2⟩2 −

2431

2∗I5⟨x − 7.75⟩2 +

∑ Δ (M

I)

w∗ ⟨x − lw⟩1

w=1:6 + ∑ (1


w=1:6 ) − (1.2753 ∗ 104)} (7)

y = 1

30∗106 {

114.975

6∗I1⟨x⟩3+

540

6∗I2⟨x − 2⟩3 −

2431

6∗I5⟨x − 7.75⟩3 +

∑ (1

2∗ Δ(M/I)w ∗ ⟨x − lw⟩2

w=1:6 ) + ∑ (1


w=1:6 ) − (1.2753 ∗ 104 ∗ x)} (8)



(b)

xz-p

lan

e

Fig

. 5

. S

hea

r fo

rce, b

end

ing

mo

men

t, d

efle

ctio

n a

nd

slo

pe

dia

gra

ms

ov

er s

haf

t le

ng

th f

or

(a)

xy-p

lan

e a

nd

(b

) xz

-pla

ne

(a)

xy-p

lan

e



Fig. 6. Plotting of (Moment/inertia) equation [M/I]

Figure 7 illustrates a comparison between the resultant slope

and deflection according to detailed and simplified analysis methods. From the comparison it is clear that the detailed

method results are more accurate, so it is implemented in

the proposed software.

Fig. 7. Slope and deflection according to detailed and simplified analysis methods

The deflection check is carried out to control deformation at

locations of power transmitting elements (gears) and

angular deformation (slope) at the location of bearings at

an acceptable range. Whether these values are acceptable or

not, it depends on the specific bearings and gears selected,

as well as the level of performance expected. The deflection check shows that all bearing slopes are well

below typical limits for bearings. The gear slopes and

deflections satisfy the recommended limits. Once

deflection or slope at a point is larger than the allowable

value, larger shaft diameter is required. A new diameter can

be found by determining the largest magnification factor

then multiplying the diameters by this ratio, consequently,

the remaining dimensions change according to suitable

ratios. Since deflection analysis has been conducted

safely within accepted ranges, there is no need for larger dimensions. Aditionaly, the FS is calculated at these

critical locations and compared with the minimum accepted

value. If the check leads to the unsafe case, the material

will be replaced by higher strength one.



4.3. Critical speed analysis

Vibration analysis is performed to check the critical speed

of the shaft. Rayleigh’s equation will be used to calculate

the first critical speed. Singularity functions method is also

used to reduce the difficulty of finding transverse

deflections of a stepped shaft caused by the weight of the shaft and the attachments. This can be performed by

dividing every (ith) shaft step into (kth) segments (except the

bearing steps). The weight of each segment (ik) is applied as

a concentrated force (fik) at the segment centroid. Also, the

gear weights are involved as concentrated forces (Wg1, Wg2)

as shown in Fig. 8. The shaft speed is compared to the

calculated critical speed to make sure that at least the shaft

is running at speed less than the half value of first critical speed. Hence, the shaft is safe; otherwise, the material

should be modified. [2, 9]

Fig. 8. Shaft loads caused by its weight and attachments

5. RESULTS AND DISCUSSION

According to the performed software, after the user enters

the required input data in the software interface shown in

Fig. 2, the user presses the CALCULATE button to start the

calculation process as described previously. The software output interface shown in Fig. 9 is then opened containing

all the results of the optimal design for the considered

stepped shaft based on the new integrated technique of

stress, deflection, and critical speed analysis. Additionally,

to aid with effective visualization, a two dimensional layout

of the optimal shaft is accomplished as shown in Fig. 10.

Also, the deflection analysis curves are plotted and can be

displayed to the user if needed.

Table 1 illustrates a comparisson between the proposed

software and Shigley design results for the implemented

case study. By comparing these results, it can be noticed that the high accuracy of the software results due to the

accuracy in determining the calculation constants compared

to the manual manner, which requires more assumptions

leading to less accuracy. Aditionaly, it is obvious that the

proposed software greatly serves any user who has no

experience or knowledge of different design techniques and

strategies, resulting in time, effort, and cost savings.

6. CONCLUSION

This paper introduces a new method based on an integrated

approach of stress, deflection and critical speed analysis to assess the optimal design of stepped shafts automatically.

The proposed software is performed with the help of

MATLAB as a programming language using a graphical

user interface and visual aids. In order to determine the

optimal shaft diameter, the fatigue stress analysis is

conducted at the most critical locations automatically.

Analysis of deflection is then performed to check whether

the resulting dimensions are safe or not. The study of

vibration is also conducted to test whether the shaft runs

safely away from the critical ones. Verification is also

carried out on the basis of a comparison between the proposed design and the Shigley design on a given case

study, which shows that the new approach used in the

proposed design leads to more accurate results. The

proposed software supports any user who has no experience

or knowledge of different design methods and strategies,

resulting in time, effort, and cost savings



Fig

. 9

. T

he o

utp

ut

inte

rfac

e o

f th

e p

rop

ose

d s

oft

war

e f

or

shaft

desi

gn



Fig. 10. A two dimensional layout of the optimal shaft

Table I

A comparisson between the proposed software and Shigley design results

D1 (in) D2 (in) D3 (in) D4 (in) D5 (in) D6 (in) D7 (in) Sut (kpsi)

Proposed software 1.13 1.35 1.62 1.94 1.62 1.35 1.13 78

Shigley design 1 1.4 1.625 2 1.625 1.4 1 100

REFERENCES

[1] Armah, S.K., Preliminary Design of a Power Transmission

Shaft under Fatigue Loading Using ASME Code. American

Journal of Engineering and Applied Sciences, 2018. 11(1): p.

227-244.

[2] Budynas, R.G. and J.K. Nisbett, Shigley's mechanical

engineering design. 9th ed. McGraw-Hill series in

mechanical engineering. 2011, New York: McGraw-Hill. xxi,

1082 p.

[3] Gujar, R. and S. Bhaskar, Shaft design under fatigue loading

by using modified Goodman method. International Journal of

Engineering Research and Applications, 2013. 3(4): p. 1061-

1066.

[4] Egelhoff, C., E. Odom, and B. Wiest. Application of modern

engineering tools in the analysis of the stepped shaft:

Teaching a structured problem-solving approach using

energy techniques. in Frontiers in Education Conference

(FIE). 2010. IEEE.

[5] Deepan Marudachalam, M., K. Kanthavel, and R. Krishnaraj,

Optimization of shaft design under fatigue loading using

Goodman method. International Journal of Scientific &

Engineering Research, 2011. 2(8): p. 1.

[6] Shivakumar, S., N.K. Anupama, and V. Khadakbhavi,

ANALYSIS OF LATHE SPINDLE USING ANSYS.

International Journal of Scientific & Engineering Research,

2013. 4(9): p. 431-440.

[7] Ravinder, S. and R. Banothu, Design and analysis of Gear

Shaft. SSRG International Journal of Mechanical Engineering

(SSRG-IJME), 2015. 2(9): p. 52-57.

[8] Tavares, S. and P. de Castro, A comparison of methodologies

for fatigue analysis of shafts: DIN 743 vs. approaches based

on Soderberg criterion. Ciência & Tecnologia dos Materiais,

2017. 29(1): p. e76-e81.

[9] Joseph E. Shigley, Charles R. Mischke, and T.H.B. Jr.,

Standard Handbook of Machine Design. 3th ed. McGraw-

Hill Professional. Vol. 1312. 2004, New York: McGraw-Hill.

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