**Introduction to Fatigue Analysis of Shafts****Shaft A Analysis**Shaft A, depicted in the figure, is crafted from AISI 1020 hot-rolled steel. It is welded to a fixed support and experiences loading by equal and opposite forces \( F \) via Shaft B. A theoretical stress-concentration factor \( K_{ts} \) of 1.6 is present due to the 1/8” fillet. The length of Shaft A from the fixed support to the connection at Shaft B measures 2 feet. The load \( F \) cycles from 150 to 500 lbf.**Key Calculations**1. **Modified Goodman Criterion** For Shaft A, the factor of safety for infinite life using the Modified Goodman fatigue failure criterion needs to be calculated.2. **Gerber Criterion** For Shaft A, calculate the factor of safety for infinite life using the Gerber fatigue failure criterion.**Diagram Description**- **Shaft A** is illustrated as a horizontal, cylindrical component fixed at its left end to a surface, with a smaller cylindrical **Shaft B** connected perpendicularly towards the right side.- **Dimensions**: - Shaft A has a diameter of 7/8 inches. - Shaft B has a diameter of 1 inch. - The fillet radius at the welding joint of Shaft A is 1/8 inch.- **Applied Forces**: - Equal and opposite forces \( F \) are applied at the end of Shaft B.These features are crucial for understanding how the design and loading conditions impact the fatigue life of Shaft A under the specified criteria.

Given data:Stress concentration factor, Fillet radius, Shaft length, Load variation, Shaft diameter, For given material, i) ultimate stress is ii) yield stress is Evaluate endurance limit modifying factors.i) ultimate strength in shear is ii) estimated endurance limit is iii) surface factor is (Here a and b are constants for hot rolled steel)iv) for , size factor is v) for torsion loading factor is vi) assume temperature factor and reliability factor are taken as 1. Thus, and .Write expression for effective endurance limit.Write expression for fatigue stress concentration factor.Here is notch sensivity.Assume shaft-B radius is same as that of fillet. For and , notch sensitivty is (obtained from standard tabular data).Thus, Write expression for torque acting on shaft A.Thus, the minimum torque is and Write expression for relation between shear stress and torque.Thus, a) minimum shear stress is b) maximum shear stress isTherefore,a) mean shear stress is b) alternating shear stress is a) Modified Goodman criteria:Write expression for modified Goodman equation.Thus, factor of saftey is 3.92.b) Gerber criteria:Write expression for Gerber equation.Solve for roots of above quadractic equation. The roots are .Since FS cannot be a negative values, Factor of safety as per modified Goodman criteria is 3.92 and as per Gerber criteria is 2.54