Bending Stresses Due to Bending Stresses Due to Stationary LoadsStationary Loads

2M

3M

θ

3

22c

3c

Iθ cosr M

Iθsin r Mσ 32

b −= Eq. 1

MMθ cos

MMθsin

MMM

3

2

23

22

=

−=

+= Eq. 3

Eq�s 4

Combining Eq�3 1,3, and 4

IrMM

σ

IrM

IrMσ

Iθ cosr M

Iθsin r Mσ

23

22

b

23

22

b

32b

⋅+−=

−−=

−=

Shaft Stress EquationsShaft Stress Equations(Bending and Torsion Only)(Bending and Torsion Only)

34x

x

πd32

d2π64Mdσ

IMcσ

M=⋅

=

=

32πd2IJ

64πdI

2dc

MMM

4

4

2z

2y

==

=

=

+=

34xy

xy

πd16T

32πd2

dTτ

JTcτ

==

=

Bending StressBending Stress Torsional Torsional StressStressReferenceReference

Principal StressesPrincipal Stresses

2xy

2xx

21 τ2σ

2σσ,σ +�

�

���

�±=

[ ]22321

2

3

2

3321

TMMπd16σ,σ

πd16T

πd16M

πd16Mσ,σ

+±=

��

���

�+��

���

�±=

Yielding in Ductile MaterialYielding in Ductile Material(No Stress Concentration)(No Stress Concentration)

( )s

yt21

2122

21 n

Sσσσσ ≤−+

Distortion Energy TheoryDistortion Energy Theory

0σ3 =

[ ]22321 TMM

πd16σ,σ +±=

Circular ShaftCircular Shaft

Distortion Energy Theory for ShaftDistortion Energy Theory for Shaft

( )s

yt3122

3 nS

3T4Mπd16 ≤+

Static Loading Minimum Static Loading Minimum DiameterDiameter

( )

31

22

yt

s

s

yt3122

3

T43M

πS32nd

nS

3T4Mπd16

��

�

�

��

�

�+=

≤+

This equation is useful in coming up with an initial shaft size early in the design process.

Shaft Stress EquationsShaft Stress Equations(Bending, Axial, and (Bending, Axial, and Torsional Torsional Loading)Loading)

( ) ( )

( )s

yt223

22321

23x

nS

48TPd8Mπd4

8Pd8MPd8Mπd2σ,σ

πd4P

πd32Mσ

≤++

���

��� ++±+=

+=

T

Axial StressAxial Stress

Principal StressesPrincipal Stresses

Distortion Energy TheoryDistortion Energy Theory

Fatigue Analysis of ShaftsFatigue Analysis of Shafts

Time

τ

Shea

r St

ress

Bending StressesBending Stresses

Torsional Torsional StressesStresses

Shaft von Shaft von Mises Mises Stress Stress EquationEquation

( )s

yt21

2122

21 n

Sσσσσ ≤−+

2xy

2xx

21 τ2σ

2σσ,σ +�

�

���

�±=

2xy

2x2

xy

2xx

2x2

1 τ2στ

2σ

2σ2

2σσ +�

�

���

�++��

���

���

���

�+��

���

�=

2xy

2x2

xy

2xx

2x2

2 τ2στ

2σ

2σ2

2σσ +�

�

���

�++��

���

���

���

�−��

���

�=

2xy

2xy

2x

2x

21 ττ2σ

2σσσ −=

��

���

��

���

+�

��

−�

��

=

Von Von Mises Mises EquationEquation

Principal Stress Principal Stress EquationEquation

s

yt2xy

2xeff n

S3τσσ ≤+=

Shaft Equivalent StressShaft Equivalent Stress

Effective Alternating and Effective Alternating and Mean StressesMean Stresses

xymeff,

xaeff,

2xy

2xeff

τ3σ

σσ

3τσσ

=

=

+= Since the bending stress is completely reversed, the only mean stress component is due to the shear stress. Since the shear stress is constant, the only alternating is equal to the maximum bending stress.

ANSI Standard Fatigue ANSI Standard Fatigue CurveCurve

A

D

B

C

E

F

ANSI Standard B106.1M-1985, �Design of Transmission Shafting�, American National Standards Institute, 1985, is based on the ASME Elliptic Fatigue Interaction Curve.

ASME Elliptic ASME Elliptic Fatigue EquationFatigue Equation

1Sσn

Sσkn

2

yt

ms

2

e

afs =��

�

�

��

�

�+��

�

����

�

This equation is used to determine whether the shaft will have infinite life. Note that the fatigue stress concentration factor has only been applied to the alternating stress. Also, the Marin factors need to be used to estimate the endurance limit.

Minimum Diameter EquationMinimum Diameter Equation(Bending and Torsion Only)(Bending and Torsion Only)

2STτ

SMσ

32πdS

3

=

=

=

1ST

43

SMk

πd32n

1ST

43

SMk

Sn

1S2S

3TnSSMkn

2

yt

2

e

f3s

2

yt

2

e

f2

s

2

yt

s

2

e

fs

=��

�

�

��

�

�+��

�

����

�

=��

�

�

�

��

�

�

��

�

�+��

�

����

���

���

�

=��

�

�

��

�

�

⋅+��

�

����

�

⋅

1Sσn

Sσkn

2

yt

ms

2

e

afs =��

�

�

��

�

�+��

�

����

�

Minimum Diameter EquationMinimum Diameter Equation(Bending and Torsion Only)(Bending and Torsion Only)

31

2

yt

2

e

fs

2

yt

2

e

f3s

ST

43

SMk

π32nd

1ST

43

SMk

πd32n

���

�

�

���

�

�

��

�

�

�+��

�

�

�=

=��

�

�

�+��

�

�

� This equation gives the minimum diameter shaft that will result in infinite fatigue life, and appears in the ANSI Standard.

Summary of Shaft Stress Summary of Shaft Stress Analysis MethodologyAnalysis Methodology

31

2

yt

2

e

fs

ST

43

SMk

π32nd

���

�

�

���

�

�

��

�

�

�+��

�

�

�=

1. Establish factor of safety that will be used in the design.2. Draw axial,shear, bending, and torsion diagrams. May

require orthogonal shear and bending diagrams.3. Determine loads acting on the shaft (M,P, and T) at

critical locations.4. Compute stress concentration factors.5. Estimate endurance limit using test data or 0.5Sut and

the Marin factors.6. Use above equation or equivalent to determine if shaft is

acceptable.

AssignmentAssignment

The shaft shown in the figure receives 110 hp from a water turbine through a chain sprocket at point C. The gear pair at E delivers 80 hp to an electrical generator. The V-belt sheave at A delivers 30 hp to a bucket elevator that carries grain to an elevated hopper. The shaft rotates at 1,700 rpm. The sprocket, sheave, and gear are located axially by retaining rings. The sheave and gear are keyed with sled runner keyseats, and there is a profile keyseat at the sprocket. The shaft is made from AISI 1040 cold-drawn steel, and has a yield strength of 71 ksi and and ultimate strength of 80 ksi. Using a design factor of safety of 3, determine the minimum diameters at each section on of the shaft.

AssignmentAssignment(Continued)(Continued)

Mott, Figure 12-13