Page 329 - Marks Calculation for Machine Design

P1: Shashi
January 4, 2005
15:4
Brown.cls
Brown˙C07
311
FATIGUE AND DYNAMIC DESIGN
7.5
COMBINED LOADING
The third type of dynamic loading to be presented is combined loading, where the total
load on the machine element is a combination of both normal (σ) and shear (τ) stresses,
whether constant, reversed, or fluctuating. The steps of the analysis to determine whether
the design is safe are as follows:
1. Calculatethe endurancelimit (S e ),except usealoadtype factor (k c = 1) for bending, and
do not apply the miscellaneous effects factor (k e ) due to the reduced stress concentration
factors (K f ) as given in Eq. (7.35).
S e = k a k b (1)k d S (7.35)
e
2. Determine the normal (σ) and shear (τ) stresses, whether constant, reversed, or fluctu-
ating, and display on a plane stress element.
3. Determine the maximum and minimum normal and shear stresses, that is, (σ max ), (σ min ),
(τ max ), and (τ min ).
4. Determine the mean and alternating normal and shear stresses, that is, (σ m ), (σ a ), (τ m ),
and (τ a ).
5. Apply any reduced stress concentration factors to the alternating stresses only, meaning
multiply (K f ) times the appropriate (σ a ) or (τ a ).
6. Multiply any alternating axial stress by (1.083 = 1/0.923) to account for the load
type factor (k c = 0.923), because the endurance limit (S e ) determined in step 1 above
assumes a load type factor for bending.
7. Use Mohr’s circle, or the applicable equations, to determine two sets of principal stresses
(σ 1 ) and (σ 2 ); one set for the mean stresses, (σ m ) and (τ m ), and the other set for the
alternating stresses, (σ a ) and (τ a ).

2

m m σ m σ m 2
σ ,σ 2 = ± + τ m (7.36)
1
2 2

2

a a σ a σ a 2
σ ,σ = ± + τ a (7.37)
2
1
2 2
8. Use the distortion-energy theory, normally used for the static design of ductile materials,
to calculate both an effective mean stress (σ eff ) and an effective alternating stress (σ eff ).
m a

eff m 2 m 2 m m

σ m = σ 1 + σ 2 − σ 1 σ 2 (7.38)

σ eff = σ a 2 + σ a 2 − σ a σ a (7.39)
a 1 2 1 2
9. Use the Goodman theory, either the mathematical equation or by plotting graphically
the appropriate stresses to determine if the design is safe. The mathematical equation
for the Goodman theory would therefore be
σ eff σ eff 1
a m
+ = (7.40)
S e S ut n
where the factor-of-safety (n) represents the distance (d) in Fig. 7.29.