How does the Goodman line differ from the Soderberg line in fatigue analysis? 

Explanation:

Fatigue Analysis: Goodman Line vs. Soderberg Line

• Fatigue analysis is essential in mechanical engineering to predict the life of a component under cyclic loading. Two common methods used in fatigue analysis are the Goodman line and the Soderberg line. These methods help engineers determine the safety and durability of materials subjected to fluctuating stresses.

Option 3: The Goodman line is based on the ultimate tensile strength, while the Soderberg line uses yield strength.

This option accurately describes the fundamental difference between the two lines used in fatigue analysis. The Goodman line incorporates the ultimate tensile strength (UTS) of the material, which is the maximum stress that a material can withstand while being stretched or pulled before breaking.
Soderberg line for ductile materials gives upper limit for any combination of mean and alternating stress. 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:

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

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:

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