 Hello students, I am Bhargay Jaishmukh from Mechanical Engineering Department, Valchan Institute of Technology, Soolapur. This session is on design against fluctuating load. Here we are going to see how to draw modified Goodman diagram for design of component which are subjected to fluctuating load. The observations are such that the mean stress component has an effect on the fatigue failure when it is present in combination with an alternating component. In the fatigue diagram, the mean stress is plotted on the abscissa, the stress amplitude is plotted on the ordinate. The magnitudes of sigma m and sigma a depends on the magnitude of maximum and minimum force acting on the component. You may think upon if mean stress is zero what is going to be the criteria for design and if amplitude stress is zero then what is the criteria for design? The modified Goodman diagram. Components under fluctuating stresses, we need to use modified Goodman diagram. Then what is the modified Goodman diagram and how to construct it? There are cases where components are subjected to fluctuating stress. Let us see the first case axial or bending stresses. If the case is axial or bending stress, you can see that the corresponding modified Goodman diagram is shown. You can see that there exists S y t, S u t on the x-axis and S e and S y t on the y-axis. As usual, sigma m or the mean stress is given on the or taken on the x-axis and amplitude stress is taken on the y-axis. However, there is a new line seen as O x e, a different line is shown. We will see what that line is, what is the significance of it. To modify the Goodman diagram, we need to use the modified Goodman diagram when the components are under fluctuating stresses. The components subjected to fluctuating stress, the case 2 is torsional shear stress. The component may be under torsional shear stress as like it is under axial or bending. The sketch, the method to draw this modified Goodman diagram differs a little bit. On the x-axis, it is S s y and S s u and on the y-axis, it is S s e, which is endurance limiting shear. Let us see the case 1, how to draw the modified Goodman diagram for the first case. A Goodman line is modified by combining fatigue failure with the failure by yielding. Let us see how to do this. In strength, S y t on the both axis, a yield line C d is drawn, which defines the failure by yielding. Mean stress as usual on the x-axis, I need to plot the stress amplitude on the y-axis. Then further, as like in the Goodman diagram, I need to put the line S y t to S y t. Let us first put S y t on the x-axis, then S y t on the y-axis, draw a line which is inclined at almost 45 degree to x-axis and y-axis. This angle is 45 degree. At the intersection on the x-axis, I need to put the point C. On the y-axis, it is point D. C d represents the line, the yielding line or it defines the controlling line for yielding. Next is S u t. I can put it on the y-axis. On y-axis, I need to put the endurance limit as per the earlier sketch also. S c to S u t, I need to join. I can get the point B as the intersection of yield line and the line joining S c and S u t. At the S c, I can mark the point as A. On the x-axis, the intersection at S u t, the point is F. If I join this line O A, A B, then B C and C to O again, if I complete this, the line A F is constructed then, we will see that line A to F. You can see over here point A and F. This line I need to join. This is constructed then. The point of intersection of these two lines, as I said, this is point B. Further, the important, highlighted by red color, the region O A, B, C represents the modified good point diagram. If I design the component to be within the stress in the component, within this zone, the component will be safe under the fatigue loading. Let us see the sketch. To solve a problem, a line O E with a slope of tan theta is constructed. The point is E. Sigma A and sigma M are used. Angle is theta. Sigma A upon sigma M, I can write it as P A upon A upon P M upon A or I can write it as amplitude force upon mean force instead of amplitude stress upon mean stress. Therefore, tan theta is also equal to P A by P M. The magnitudes of P A and P M can be determined from the maximum and minimum forces acting on the component. Tan theta is also equal to M B A upon M B M, where M B A is amplitude bending moment and M B M represents the mean bending moment. M B A is the amplitude bending moment. M B M is the mean bending moment. Intersections of A B and O E is X. Here the point X is defined, which indicates dividing line between the safe region and the region of failure. I am marking the points S A. Horizontal line from X will give me S A. Vertical line from X, which is cutting, intersecting the X axis, I can get S M. The coordinates of X, which are S M and S A, represents the limiting values of the stresses used to design the component. The permissible stress here is sigma A, which is S A upon F S and sigma M, which is equal to S M upon F S. It is to be noticed over here that S A is the Y coordinate of point X and S M is the X coordinate of point X. The process has been carried out. You can recall the process. We have initially plotted the point S Y T on the X axis, S Y T on the Y axis. Then we have plotted the point S E on the Y axis. We have joined this point S E to the S U T. At the intersection of this line and this E line, I can get the point B, which is the intersection of these two. And then I can see that point A B C then O to A. This is the region of safety that we have obtained. Further, it was seen that this theta, the angle, how to get that angle theta? For this particular case of axial or bending stress, sigma A by sigma M, M B A by M B M, P A by P M, all the equations we can use and get the angle theta. And ultimately, the permissible stresses can be obtained by sigma A equals S A upon F S and sigma M equals S M upon F S. For the torsional or the fluctuating torsional shear stress, the diagram can be obtained as shown. The only change is S S E is joined to S S U and from S S E, I need to draw the horizontal line. S S Y is not extended on the Y axis. Line A B is horizontal in that case. The torsional mean stress on the abscissa, torsional stress amplitude on the ordinate I need to put. And then the torsional yield strength S S Y is plotted on the abscissa and the yield line is constructed, which is inclined at 45 degree to the abscissa. You can please check this. It is interesting to note that up to a certain point, the torsional mean stress has no effect on the torsional endurance limit. A line is drawn through S S E on the ordinate and parallel to the abscissa. You can see that this is the line A to B, which is drawn parallel to the X axis on the sketch over here. A B is parallel to the X axis. The point of intersection of this line and the yield line is B line, horizontal line and the yield line I can get the point B. Area O A B C represents the region of safety here also. It is not necessary to construct a fatigue diagram for fluctuating torsional shear stresses because A B is parallel to the X axis. Instead, a fatigue failure is indicated if tau A equals S S E. A static failure is indicated if tau max equals tau A plus tau M that equals S S Y or yield strength in shear. The permissible stress tau A equals S S E upon F S and tau max equals S S Y upon F S.