Site pages
Current course
Participants
General
MODULE 1.
MODULE 2.
MODULE 3.
MODULE 4.
MODULE 5.
MODULE 6.
MODULE 7.
MODULE 8.
MODULE 9.
MODULE 10.
LESSON 7. DESIGN FOR DYNAMIC LOADING  II
7.1 Design for Completely Reversed Stresses
As discussed earlier, in case of completely reversed stress cycles, extreme values of stress are of equal magnitude and opposite nature with mean equal to zero. Design problems for completely reversed stresses can be divided into two groups:
i. Design for Infinite Life ii. Design for Finite Life
7.1.1 Design for Infinite Life
If the stress developed in a component is kept below the endurance limit, it can survive for infinite number of cycles or can have infinite life. Thus endurance limit is the design criteria in this case and the design equation can be written as:
7.1.2 Design for Finite Life
When components are designed to survive for 10^{3} to 10^{6} number of cycles, it is called design for finite life. For SN curve of steel shown in figure 7.1, line AB represents this region.
To design for finite life, fatigue strength is taken as design criteria. Fatigue strength for required number of stress cycles can be determined graphically from the SN curve or mathematically by using the equation of line AB. Or alternatively, for a known value of induced stress, number of cycles that the component will survive can be calculated. SN curve is a loglog plot and coordinates of points A and B are:
\[A(3,lo{g_{10}}\left( {0.9{S_{ut}}} \right)andB\left( {6,lo{g_{10}}{S_e}} \right)\]
Ordinate of point A is approximately 90% of the ultimate tensile strength.
Equation of line AB can be written as,
\[lo{g_{10}}\left( {{S_f}} \right) = blo{g_{10}}N + c\] or \[{S_f} = {N^b}{10^c}\]
Where, b and c are constants that can be determined using coordinates of A and B.
\[b = \frac{1}{3}lo{g_{10}}\left( {\frac{{{S_e}}}{{0.9{S_{ut}}}}} \right)andc = lo{g_{10}}\left( {\frac{{{{(0.9{S_{ut}})}^2}}}{{{S_e}}}} \right)\]
7.2 Design for Fluctuating Stresses
In this case, mean value of stress has a nonzero value and both static and variable components of stress contribute to the failure. Figure 7.2 shows the scatter of failure points obtained from various experiments performed with different combinations of σ_{m} and σ_{a}. Observing these results, Soderberg, Goodman and Gerber have proposed three different theories, defining the boundary line between the safe and unsafe region on the σ_{m} vs σ_{a} plot. Soderberg line, Goodman Line and Gerber Parabola are described in table 7.1 and are shown in figure 7.2. When stress amplitude (_{σa}) is zero, stress is purely static and S_{yt} or S_{ut} are the criteria of failure. These are plotted on the abscissa, along which s_{m} is plotted. When the mean stress (σ_{m}) is zero, stress is completely reversing and S_{e} is the criteria of failure. It is plotted on the ordinate, along which σ_{a} is plotted.
Table 7.1 Design for Fluctuating Stresses
Criteria 
Soderberg Line 
Goodman Line 
Gerber Parbola 
Description 
Line joining S_{e} on the ordinate to S_{yt} on the abscissa. 
Line joining S_{e} on the ordinate to S_{ut} on the abscissa. 
Parabolic curve joining S_{e} on the ordinate to S_{ut} on the abscissa. 
Equations 
\[\frac{{{\sigma _m}}}{{{S_{yt}}}} + \frac{{{\sigma _a}}}{{{S_e}}} = 1\] 
\[\frac{{{\sigma _m}}}{{{S_{ut}}}} + \frac{{{\sigma _a}}}{{{S_e}}} = 1\] 
\[{\left( {\frac{{{\sigma _m}}}{{{S_{ut}}}}} \right)^2} + \frac{{{\sigma _a}}}{{{S_e}}} = 1\] 
Considering Factor of Safety 
\[\frac{{{\sigma _m}}}{{{S_{yt}}/fos}} + \frac{{{\sigma _a}}}{{{S_e}/fos}} = 1\] 
\[\frac{{{\sigma _m}}}{{{S_{ut}}/fos}} + \frac{{{\sigma _a}}}{{{S_e}/fos}} = 1\] 
\[{\left( {\frac{{{\sigma _m}}}{{{S_{ut}}/fos}}} \right)^2} + \frac{{{\sigma _a}}}{{{S_e}/fos}} = 1\] 
Final Design Equation 
\[\frac{{{\sigma _m}}}{{{S_{yt}}}} + \frac{{{\sigma _a}}}{{{S_e}}} = \frac{1}{{fos}}\] 
\[\frac{{{\sigma _m}}}{{{S_{ut}}}} + \frac{{{\sigma _a}}}{{{S_e}}} = \frac{1}{{fos}}\] 
\[{\left( {\frac{{{\sigma _m}}}{{{S_{ut}}}}} \right)^2}.fos + \frac{{{\sigma _a}}}{{{S_e}}} = \frac{1}{{fos}}\] 
As discussed in the case of static loading, allowable stress or design stress is obtained by dividing S_{ut} or S_{yt} by factor safety. Similarly, in the present case, taking factor of safety reduces the safe region as shown in figure 7.3 for Soderberg line.
References

Mechanical Engineering Design by J.E. Shigley

Analysis and Design of Machine Elements by V.K. Jadon

Design of Machine Elements by VB Bhandari

Design of Machine Elements by M. F. Spotts

Design of Machine Elements by C.S. Sharma & K. Purohit

Machine Elements in Mechanical Design by Robert L. Mott