Square head for shaft-hub connection

The advantage of this connection is easy assembly and disassembly. The disadvantage is the low manufacturing precision and consequent consequences for limited speeds and small torques.

For a simplified calculation, it is assumed that the joint is without will, and that the torque causes the contact stress to be half of each function area of the square head. It is possible to assume a triangular distribution of this stress.

Load distribution will differ from the assumption due to production inaccuracy due to looseness or prestressing of joints and shaft deformations by from torsion torque. These deviations can include in the calculation a coefficient max. stress Ss=1,3-2 the lower value of which applies to short joints l≤s and for high accuracy of manufacturing.

Fig.1 square head for shaft-hub connection

Bearing stress:

p=MT*ss2a*l*bσall

p - bearing stress - [MPa]

MT - torque - [Nm]

ss - coefficient of maximum stress increase - []

a - length square head load - [mm]

l - length square head in the hub - [mm]

b - distance of the resultant of the pressure - [mm]


Distance of the resultant of the pressure:

b=a1+23a

b - distance of the resultant of the pressure - [mm]

a1 - length square head without load - [mm]

a - length square head load - [mm]


Length square head without load:

a1=d92sincos-1sd9

a1 - length square head without load - [mm]

d9 - free diameter - [mm]

s - width square head - [mm]


Length square head with load:

a=d82sincos-1sd8-a1

a - length square head with load - [mm]

d8 - diameter square head - [mm]

s - width square head - [mm]

a1 - length square head without load - [mm]


Allowable bearing stress:

σall=0,9Rp0,2TSF*Cc

σall - allowable bearing stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []


Coefficient of use of joints according to load:

load[]
Unidirectional load, non-impact load0,8
Unidirectional load, with a small impact load0,7
Unidirectional load, with a big impact load0,6
Alternating load, with a small impact load0,45
Alternating load, with a big impact load0,25

Torsion stress in the shaft:

τs=16MTπd3τall

τs - torsion stress in the shaft - [MPa]

MT - torque - [Nm]

d - diameter of the shaft - [mm]

τall - allowable shear stress - [MPa]


Allowable shear stress:

τall=0,4Rp0,2TSF*Cc

τall - allowable shear stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []


Torsion stress in the hub:

τh=3,96216MTπDh4-4s4/Dhτall

τh - torsion stress in the hub - [MPa]

MT - torque - [Nm]

Dh - diameter of the hub - [mm]

s - width square head - [mm]

τall - allowable shear stress - [MPa]

If the shaft is loaded with the bending moment in the joint, the bending stress must be checked. If the shaft is loaded with a shear force in the joint, the shear stress must be checked. The shaft may be load in the joint by axial force. The shaft must be checked for axial stresses. When calculating the different load types, it is necessary to calculate the combined stress.


Bending stress in the shaft:

σB=32MBπd3σBall

σB - bending stress in the shaft - [MPa]

MB - bending moment - [Nm]

d - diameter of the shaft - [mm]

σBall - allowable bending stress - [MPa]


Allowable bending stress:

σBall=0,6Rp0,2TSF*Cc

σBall - allowable bending stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []


Shear stress in the shaft:

τss=4FRπd2τall

τs(s) - shear stress in the shaft - [MPa]

FR - shear forces - [N]

d - diameter of the shaft - [mm]

τall - allowable shear stress - [MPa]


Axial stress in the shaft:

σA=4FAπd2σAall

σA - axial stress in the shaft - [MPa]

FA - axial forces - [N]

d - diameter of the shaft - [mm]

σAall - allowable axial stress - [MPa]


Allowable axial stress:

σAall=0,45Rp0,2TSF*Cc

σAall - allowable axial stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []


Combined stress in the shaft:

σtresca= KtB*σB2+KtA*σA2+4Kts*τs2+τss2 σCall

σtresca - combined stress in the shaft - [MPa]

KtB - concentration factor in bending stress - []

σB - bending stress in the shaft - [MPa]

KtA - concentration factor in axial stress - []

σA - axial stress in the shaft - [MPa]

Kts - concentration factor in torsion stress - []

τs - torsion stress in the shaft - [MPa]

τs(s) - shear stress in the shaft - [MPa]

σCall - allowable combined stress - [MPa]


Concentration factor in bending stress:

KtB=C1B+C2BD-dD +C3BD-dD2+C4BD-dD3
C1B=0,947+1,206D-d2r -0,131D-d2r
C2B=0,022-3,405D-d2r +0,915D-d2r
C3B=0,869+1,777D-d2r -0,555D-d2r
C4B=-0,810+0,422D-d2r -0,260D-d2r

KtB - concentration factor in bending stress - []

C1B - coefficient - []

C2B - coefficient - []

C3B - coefficient - []

C4B - coefficient - []

D - diameter of the shaft - [mm]

d - diameter of the shaft - [mm]

r - radius - [mm]


Concentration factor in axial stress:

KtA=C1A+C2AD-dD +C3AD-dD2+C4AD-dD3
C1A=0,926+1,157D-d2r -0,099D-d2r
C2A=0,012-3,036D-d2r +0,961D-d2r
C3A=-0,302+3,977D-d2r -1,744D-d2r
C4A=0,365-2,098D-d2r +0,878D-d2r

KtA - concentration factor in axial stress - []

C1A - coefficient - []

C2A - coefficient - []

C3A - coefficient - []

C4A - coefficient - []

D - diameter of the shaft - [mm]

d - diameter of the shaft - [mm]

r - radius - [mm]


Concentration factor in torsion stress:

Kts=C1s+C2sD-dD +C3sD-dD2+C4sD-dD3
C1s=0,905+0,783D-d2r -0,075D-d2r
C2s=-0,437-1,969D-d2r +0,553D-d2r
C3s=1,557+1,073D-d2r -0,578D-d2r
C4s=-1,061+0,171D-d2r +0,086D-d2r

Kts - concentration factor in torsion stress - []

C1s - coefficient - []

C2s - coefficient - []

C3s - coefficient - []

C4s - coefficient - []

D - diameter of the shaft - [mm]

d - diameter of the shaft - [mm]

r - radius - [mm]


Allowable combined stress:

σCall=Rp0,2TSF*Cc

σCall - allowable combined stress - [MPa]

Rp0,2T - the minimum yield strength or 0,2% proof strength at calculation temperature - [MPa]

SF - safety factor - []

Cc - coefficient of use of joints according to load - []


Literature:

AISC: Specification for structural steel buildings: Allowable Stress design and plastic design 1989

Walter D. Pilkey, Deborah F. Pilkey: Peterson’s stress concentration factors. 2008

Joseph E. Shigley, Charles R. Mischke, Richard G. Budynas: Konstruování strojních součástí 2010.

MET-Calc: Allowable stress

A. Bolek, J. Kochman a kol.: Části a mechanismy strojů I. 1989.

K. Kříž a kol.: Strojní součásti 1. 1984.


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Square head for shaft-hub connection.pdf


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