Page 288 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
266
(3) Determine the transmitted torque and develop a torque diagram, usually between
the midpoint of torque transmission elements, see Figure 10.3e;
(4) After the bending moment and torque have been decided, the maximum shear stress
theory can be used to calculate design stress by
√
2
= + 4 2
ca
Since the bending stress in the shaft is completely reversed stress as shafts rotate while
torsional shear stress is usually not, a correction coefficient is introduced to consider
the difference in stress ratios between bending and torsional shear stresses. The design
stress is modified as
√
= + 4( ) 2 (10.3)
2
ca
If torsional stress is a static stress, select = 0.3; if torsional stress is repeated stress
or unknown, select = 0.6; if torsional stress is also a completely reversed stress, select
= l.0. For circular shafts, the bending stress is [5]
M
= (10.4)
W
and torsional shear stress is [5]
T T
= = (10.5)
W T 2W
where W and W are section modulus and polar section modulus, respectively.
T

Substitute Eqs. (10.4) and (10.5) to Eq. (10.3), the design stress for rotating, solid cir-
cular shafts at the critical cross section is
√
M ca M +(aT) 2
2
= = ≤ [ ] (10.6)
ca
−1
W 0.1d 3
where M is the equivalent bending moment, and allowable bending stress [ ]can be
−1
ca
found in Table 10.1.
Since both loads and cross sections vary along the shaft, it is necessary to calculate
and evaluate stresses at several critical sections. Possible critical locations of maximum
stress may be decided by a combined effect of diameter variation and high torque and
bending moments.
When a shaft is subjected to bending only, that is, an axle, no torque is applied.
Substitute T = 0 in Eq. (10.6) so the stress can then be calculated.
10.3.1.3 Fatigue Strength Analysis
To evaluate fatigue strength of a shaft, stress variation, diameter changes, stress concen-
trations and surface conditions of shaft must be taken into account. The fatigue strength
is usually calculated at critical locations on the outer surface, where the stress magni-
tude is large and where stress concentrations exist [2]. Normally, several critical sections
along a stepped shaft are selected, and the stresses at each critical point are calculated
and compared with the allowable values to ensure safety. The fatigue strength analysis
of combined stresses introduced in Eq. (2.50) is duplicated here as
S S

ca
S = √ ≥ [S] (10.7)
2
S + S 2