This document discusses cumulative fatigue damage and design for fluctuating loads. It introduces Miner's rule for cumulative damage calculation and modified Goodman diagrams. For components under fluctuating axial/bending stresses, the modified Goodman diagram uses a line with slope tanθ intersecting the endurance limit and yield lines to divide the safe and failure regions. For torsional shear stresses, the safe region is above a horizontal line through the endurance limit and below the yield line.

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 (σ₁) for (n₁)
cycles, (σ₂) for (n₂) cycles, and so on. Let N₁ be the number of stress cycles before fatigue
failure, if only the alternating stress (σ₁) 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
(σ₂) 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 σ₁, σ₂, ... are unknown. Suppose
that α₁, α₂, ... are proportions of the total life that will be consumed by the stress levels σ₁,
σ₂,... etc. Let N be the total life of the component. Then,
n₁ = α₁ N
n2 = 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 (σm) as well as
stress amplitude (σa). 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
(σm) and (σa) stress depend upon the maximum and minimum force acting on the
component. When stress amplitude (σa) is zero, the load is purely static and the criterion of
failure is Sut or Syt. These limits are plotted on the abscissa. When the mean stress (σm) is
zero, the stress is completely reversing and the criterion of failure is the endurance limit
(Se) that is plotted on the ordinate. When the component is subjected to both components
of stress, viz., (σm) and (σa), 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 (σm) and (σa). Different criterions are proposed to construct the borderline
dividing safe zone and failure zone.

5. • Gerber Line: A parabolic curve joining Se on the ordinate to Sut on the abscissa is called the
Gerber line.
• Soderberg Line: A straight line joining Se on the ordinate to Syt, on the abscissa is called the
Soderberg line.
• Goodman Line: A straight line joining Se on the ordinate to Sut on the abscissa is called the
Goodman line.
• First Cycle of stress: A yield line is constructed (Syt) 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:
𝑥
𝑎
+
𝑦
𝑏
= 1
• Where, a & b are intercepts of the line on the X and Y axes respectively.
• For Soderberg line :
σm
Syt
+
σa
S𝑒
= 1
• For Goodman line:
σm
Sut
+
σa
S𝑒
= 1

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:
1. The Goodman line is safe from design consideration because
failure points are outside of it.
2. 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 (Mb)m & (Mb)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 (Sm, Sa)
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 (Sm, Sa) 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 Sse 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: