# Static Failure Theories

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```					Shaft Design Considerations
Design Considerations:

 to minimize both shaft deflections and stresses, shafts should
be as short as possible

 cantilevers are not recommended unless required for

 avoid stress concentrations at regions of high bending
moments; minimize their effect through generous reliefs

S. Waldman                         MECH 323
Shaft Design Constraints
Design Constraints:

 deflections at gears should not exceed: 0.1 mm

 relative slopes between mating gear axes should be held to
less than: 0.0005 rad

 shaft slope at bearings should be kept less than:
•   cylindrical roller bearings:   0.0001 rad
•   tapered roller bearings:       0.0005 rad
•   deep-groove roller bearings:   0.004 rad
•   spherical ball bearings:       0.0087 rad

S. Waldman                             MECH 323
What happens if there is loading in more than one plane (i.e.
vertical and horizontal planes)?

 the simplest method is to create separate bending moment
diagrams for each plane and then determine the resultant
moment at sites of interest using Pythagorem’s theorem:

M resultant  M vertical  M horizontal
2                2

 the same procedure should be used for determining shaft
slopes and deflections:

tan  resultant  tan 2  vertical  tan 2  horizontal

 resultant   vertical2   horizontal 2

S. Waldman                                 MECH 323
Design for Fluctuating Bending and Torsion
The von Mises stress amplitude component sa´ and mean
component sm´ are given by:

4K f M a   3K fsTa   3
16                               16 A
s a  s x,a  3 xy,a 
     2        2                        2            2

d 3                             d
 3 4K f M m   3K fsTm   3
16                               16B
s m  s x ,m  3 xy,m
     2         2                         2            2

d                               d

where A and B are the radicals in the above equations.

The Gerber fatigue failure criterion is defined by:

2                    2                     2
Sa  Sm        ns a  ns m 
         16nA  16nB            
                     3  3                 1
S e  Sut


    S e  Sut  d S e  d Sut
          



S. Waldman                                 MECH 323
Design for Fluctuating Bending and Torsion
solving for the shaft diameter d:
1
                               1
       3
 8nA    2 BSe                      
2          2

d                  AS  
1  1                        
  Se            ut                
                                     

or, solving for the factor of safety, n:

                     1
2
 2 BSe   
2
1   8A 
                 AS   
1  1         
n  d Se   
3
ut  
                      
where:
A  4K f M a   3K fsTa 
2              2

B  4K f M m   3K fsTm 
2              2

S. Waldman                          MECH 323
Determining Shaft Deflections
Various method exist to determine the deflections of beams due to
bending.

The complicating factor for the design of shafts is typically the
presence of step changes in shaft diameter along its length
(shoulders, etc.). Thus, one commonly used method is the
Integration Method with aid of Singularity Functions.

d4y
distributed load function:          q  EI 4
dx
d3y
shear force function:               V  EI 3
dx
d2y
moment function:                    M  EI 2
dx
dy
slope function:                     
dx
S. Waldman                        MECH 323
Determining Shaft Deflections
Integrating:
V   qdx  C1

M   Vdx  C1 x  C2

M
       dx  C1 x 2  C2 x  C3
EI
y    dx  C1 x3  C2 x 2  C3 x  C4

The integration constants C1 and C2 are the boundary conditions
on the shear and moment function, which are simply the reaction
forces imposed on the beam.

Thus, if the reaction forces are used in the analysis (which is a very
good idea):
C1  C2  0

S. Waldman                          MECH 323
Singularity Functions
Singularity functions are use to represent discrete entities (loads,
moments, etc.) applied in a discontinuous fashion over the beam
length.

Denoted by the binominal function in angled brackets:
xa
n

where:       x is the variable of interest
a is a user defined parameter to denote where in x the
singularity acts
n is the power of the function

S. Waldman                        MECH 323
Singularity Functions
Commonly used functions are:

f  x  a : f  0 for x  a ; f  x  a  for x  a
2                                 2
unit parabolic function:

f  x  a : f  0 for x  a ; f  x  a for x  a
1
unit ramp function:

f  x  a : f  0 for x  a ; f  1 for x  a
0
unit step function:
1
unit impulse function:         f  xa              : f  0 for x  a ; f  0 for x  a

Integration of Singularity Functions:

1                 n 1
   x  a dx         xa                   , n0
n

n 1

n 1
 xa           ,               n0
S. Waldman                          MECH 323
Torsional Deflection
Angular deflection of a shaft from torsional loads is:
Tl

GJ

where: T is the torque
l is shaft length
G is the shear modulus
J is the polar moment

If the shaft is stepped or has multiple torques applied to it, the
angular deflection can be determined from the sum of the
deflections of each shaft segment:
Ti li
               where,      i is the shaft segment
Gi J i

S. Waldman                      MECH 323
Example

S. Waldman   MECH 323

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