Which one of the following failure theories is the most conservative design approach against fatigue failure?

Explanation:

Soderberg line:
Soderberg line for ductile materials gives upper limit for any combination of mean and alternating stress. Soderberg line is the most conservative fatigue failure criterion. Following diagram depicts the same.

Considering following notations

σa = limiting safe stress amplitude

Se = endurance limit of the component

σm = limiting safe mean stress

Sut = ultimate tensile strength

Syt = Yield strength

N = Factor of safety

Soderberg line:

The line joining Syt (yield strength of the material) on the mean stress axis and Se (endurance limit of the component) on stress amplitude axis is called as Soderberg line. This line is used when yielding defines failure (Ductile materials).

The equation for the Soderberg line:

\(\frac{{{{\rm{\sigma }}_{\rm{m}}}}}{{{{\rm{S}}_{{\rm{yt}}}}}} + \frac{{{{\rm{\sigma }}_{\rm{a}}}}}{{{{\rm{S}}_{\rm{e}}}}} = \frac{1}{N}\)

Additional Information

Goodman line:

Line joining Se on stress amplitude axis and Sut on mean stress axis is known as Goodman line. The triangular region below this line is considered a safe region. 
The equation for Goodman line:

\(\frac{{{\sigma _m}}}{{{S_{ut}}}} + \frac{{{\sigma _a}}}{{{S_e}}} = \frac{1}{N}\)

Gerber Line: 

Line joining Se on stress amplitude axis and Sut on mean stress axis is joined by a parabolic curve. 
The equation for Gerber line: