Design against fluctuating load

Machine design

1. Design of Machine Elements Design against fluctuating load Lecture: 4 Presented By: Jagdip Chauhan Assistant Professor MED, GJUS&T, Hisar

2. CUMULATIVE DAMAGE IN FATIGUE: In certain applications, the mechanical component is subjected to different stress levels for different parts of the work cycle. The life of such a component is determined by Miner's equation. Suppose that a component is subjected to completely reversed stresses ( σ ₁) for (n₁) cycles, ( σ ₂) for (n₂) cycles, and so on. Let N₁ be the number of stress cycles before fatigue failure, if only the alternating stress ( σ ₁) is acting. One stress cycle will consume (1/N₁) of the fatigue life and since there are n₁ such cycles at this stress level, the proportionate damage of fatigue life will be [(1/N₁)n₁] or (n₁/N₁). Similarly, the proportionate damage at stress level ( σ ₂) will be (n₂/N₂). Adding these quantities, we get (Miner’s equation)

3. CUMULATIVE DAMAGE IN FATIGUE: Sometimes, the number of cycles n₁, n₂, ... at stress levels σ ₁, σ ₂, ... are unknown. Suppose that α₁, α₂, ... are proportions of the total life that will be consumed by the stress levels σ ₁, σ ₂,... etc. Let N be the total life of the component. Then, n₁ = α₁ N n 2 = α₂ N Substituting these values in Miner's equation,

4. Design for fluctuating stresses: When a component is subjected to fluctuating stresses, there is mean stress ( σ m ) as well as stress amplitude ( σ a ). It has been observed that the mean stress component has an effect on fatigue failure when it is present in combination with an alternating component. The fatigue diagram for this general case is shown by the diagram, in which the mean stress is plotted on the abscissa. The stress amplitude is plotted on the ordinate. The magnitudes of ( σ m ) and ( σ a ) stress dep end upon the maximum and minimum force acting on the component. When stress amplitude ( σ a ) is zero, the load is purely static and the criterion of failure is Sut or S yt . These limits are plotted on the abscissa. When the mean stress ( σ m ) is zero, the stress is completely reversing and the criterion of failure is the endurance limit (S e ) that is plotted on the ordinate. When the component is subjected to both components of stress, viz., ( σ m ) and ( σ a ), the actual failure occurs at different scattered points shown in the figure. There exists a border, which divides safe region from unsafe region for various combinations of ( σ m ) and ( σ a ). Different criterions are proposed to construct the borderline dividing safe zone and failure zone.

5. Gerber Line: A parabolic curve joining S e on the ordinate to S ut on the abscissa is called the Gerber line. Soderberg Line: A straight line joining S e on the ordinate to S yt , on the abscissa is called the Soderberg line. Goodman Line: A straight line joining S e on the ordinate to S ut on the abscissa is called the Goodman line. First Cycle of stress: A yield line is constructed ( S yt ) on both the axes. It is called as first cycle of stress.

6. Design for fluctuating stresses: Soderberg and Goodman lines are straight lines. Hence, these are solved by straight line equation: Where, a & b are intercepts of the line on the X and Y axes respectively. For Soderberg line : For Goodman line:  

7. Design for fluctuating stresses: Goodman line is generally used as the criterion of fatigue failure when the component is subjected to fluctuating stress, because: The Goodman line is safe from design consideration because failure points are outside of it. It follows the straight line equation which is much simpler than parabolic curve.

8. Modified Goodman Diagram: The components, which are subjected to fluctuating stresses, are designed by constructing the modified Goodman diagram. For the purpose of design, the problems are classified into two groups: ( i ) components subjected to fluctuating axial or bending stresses; and  (ii) components subjected to fluctuating torsional shear stresses. Separate diagrams are used in these two cases.

9. Modified Goodman Diagram: The modified Goodman diagram for fluctuating axial or bending stresses

10. Modified Goodman Diagram: The modified Goodman diagram for fluctuating axial or bending stresses

11. For solving the line OE with a slope of tan Ɵ is constructed in such a way that, The magnitude of Pa and Pm can be determined from maximum and minimum forces acting on the component. Similarly, it can be proved that, The magnitude of (M b ) m & (M b ) a can be determined from maximum and minimum bending moment acting on the component.

12. The point of intersection of line AB & OE is X. The point X indicates the deviding line between the safe region and the region of failure.The coordinates of the point X ( S m , S a ) represent the limiting values of stresses, which are used to calculate the dimensions of the component. The permissible stresses are as follows: Modified Goodman Diagram:

13. The point of intersection of line AB & OE is X. The point X indicates the deviding line between the safe region and the region of failure.The coordinates of the point X ( S m , S a ) represent the limiting values of stresses, which are used to calculate the dimensions of the component. The permissible stresses are as follows: Modified Goodman Diagram:

14. Modified Goodman Diagram: The modified Goodman diagram for fluctuating torsional shear stresses

15. The torsional mean stress shows no effect on the endurance limit after a certain point. Therefore , a line is drawn through S se on the ordinate and parallel to the abscissa. The point of intersection of this line and the yield line is B. The area OABC represents the region of safety. It is not necessary to construct a fatigue diagram for fluctuating torsional shear stresses because AB is parallel to X-axis. Fatigue failure is given by:

16. Thanks