Page 307 - Marks Calculation for Machine Design
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P1: Shashi
                          January 4, 2005
                                      15:4
                 Brown˙C07
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                                        FATIGUE AND DYNAMIC DESIGN
                                             Yield line                           289
                               Alternating stress (s a )  S e  Soderberg line  Gerber line
                                 S
                                  y
                                                                Goodman line
                                  0
                                    0               S y      S ut
                                                Mean stress (s )
                                                           m
                              FIGURE 7.11  Goodman, Soderberg, and Gerber lines.


                      The mathematical expression for the Soderberg theory is given in Eq. (7.21),
                                               S a  S m
                                                  +    = 1                      (7.21)
                                               S e  S y
                    the mathematical expression for the Goodman theory is given in Eq. (7.22),
                                               S a  S m
                                                  +    = 1                      (7.22)
                                               S e  S ut
                    and the mathematical expression for the Gerber theory is given in Eq. (7.23),
                                                       2

                                              S a  S m
                                                +       = 1                     (7.23)
                                              S e  S ut
                    where (S m ) is the mean strength and (S a ) is the alternating strength.
                      Factors-of-safety (n) can be established for each of these three theories by substituting
                    the actual mean stress (σ m ) for the mean strength (S m ), substituting the actual alternating
                    stress (σ a ) for the alternating strength (S a ), and substituting (1/n) for 1.
                      For the Soderberg theory the factor-of-safety (n) is found from Eq. (7.24),
                                               σ a  σ m  1
                                                 +    =                         (7.24)
                                               S e  S y  n
                    for the Goodman theory the factor-of-safety (n) is found from Eq. (7.25),

                                               σ a  σ m  1
                                                 +    =                         (7.25)
                                               S e  S ut  n
                    and for the Gerber theory the factor-of-safety is found from Eq. (7.26),
                                                        2

                                             nσ a  nσ m
                                                +        = 1                    (7.26)
                                             S e    S ut
                      As the Goodman theory is the most commonly used, it is shown by itself in Fig. 7.12
                    where the mean stress (σ m ) and alternating stress (σ a ) are plotted.
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