 Hello students, I am Bhagyash Deshmukh from Mechanical Engineering Department, Walsh and Institute of Technology, Solapur. This session is on Sodorberg and Goodman lines. Consider the design of components subjected to fluctuating or alternating stresses. Can you think what is static loading and what could be the fluctuating load? How these two cases are going to differ? What is the design criteria for static loading? What is the design criteria for fluctuating load? Let us see this stress cycle. Here you can see that on the x-axis it is the time scale and on the y-axis it is the magnitude of the stress. Here the stress is only shown on the plus end. There can be similar stress cycle on the minus side. However, for this case we have just taken the component under stress on only tensile end or the positive end. There exists a sigma max which is the maximum stress, sigma mean which is the mean stress m e a n mean, sigma minimum which is indicated as sigma min and there exists sigma a which is the amplitude stress. For these kind of designs sigma a plays very important role. Let us see why this fluctuation is dangerous. The fluctuation the load the stress varies in a sinusoidal manner with respect to time. It has some mean value as well as the amplitude value. It fluctuates between two limits. The upper limit is sigma max and the lower limit is the sigma min or the minimum stress. As I said the stress can be tensile or compressive in nature. This is how the stress cycle is seen with respect to time it varies as a sinusoidal wave. We can say that for this typical case of fluctuating stress the value of mean stress is not equal to 0 and the value of amplitude stress is not equal to 0. If you call the reverse stress the mean stress will be equal to 0 in that case but this typical case deviates from that case as sigma min and sigma amplitude exist. Design of these kind of components we need to check how to design it then. The observation is that the mean stress has a component which has a effect on the fatigue failure when it is present in combination with an alternating component. Mean as well as amplitude it has effect on the design. Now we are drawn this fatigue diagram in this fatigue diagram the mean stress is plotted on the abscissa and the stress amplitude is plotted on the ordinate. The magnitudes of sigma m and sigma a depend upon the magnitudes of maximum and minimum pores acting on the component. Now the mean stress is taken on the x-axis and the amplitude stress is taken on the y-axis syt and syt we need to plot on the both axis and draw a line from these two. If I draw this line the angle made by this line with respect to the x-axis and y-axis both it is 45 degree I can say that this is the yield line or it is the criteria failure under the static load. The component may fail by yielding I need to put the sut value on the x-axis and se value on the y-axis. I need to join the line from se to syt I need to put this line as the Sodorberg line then I need to mention what is the Sodorberg line the Sodorberg line is the line which joins the endurance limit se on the ordinate or the y-axis to the syt on the abscissa or the x-axis this line is called as the Sodorberg line you can see that this line is well below the yield line I need to join the se with sut if I join the se to sut then this line is called as the Goodman line a Goodman line is a straight line which joins se on the ordinate or the y-axis on the abscissa or the x-axis this line is called as a Goodman line I can draw another curve which joins se to sut I can plot the points this Gerber line it is a parabolic curve which joins se on the ordinate to the sut on the abscissa and is called as the Gerber line now you can see that on the x-axis it is the mean stress when the stress amplitude sigma a is 0 where students you need to note that sigma a is 0 what will be the case if it is sigma a is 0 I can say that it is the pure static loading case the criteria of failure is sut or syt in that case we have taken the stress amplitude on the y-axis I can say that if for the static loading this value is going to be equal to 0 and y-axis hence is not considered the stress amplitude is not there and the criteria for design is sut or syt these limits sut and syt are then plotted on the abscissa then in the next case where the mean stress is 0 the stress is completely reversing and the criteria of the failure is endurance limit se that is plotted on the ordinate or the y-axis this is for the design of components subjected to fluctuating or alternating stresses let us see how it happens on the x-axis it is mean stress and on the y-axis it is the amplitude stress or the stress amplitude I said that the mean stress is 0 if the load is completely reverse or the stress is completely reverse and the criteria of failure is purely se x-axis is now gone se is the criteria for design but when these two cases exist that means there is a stress amplitude on the y-axis there is the stress mean stress on the x-axis the both components are existing together the case is of the loading in such a way that there exists a mean stress there exists an amplitude stress then how to design it this is a very typical case of the design and where I cannot design the component on the basis of syt because the component is not purely under the static loading syt syt I cannot take that line the yield line will not be the criteria for design then how to design these components then the lines Soderbergh line Goodman line Gerber lines these lines can be used you can see that there are the failure points located and the Gerber lines almost matches or the Gerber lines passes these these through these failure points when the component is subjected to these both components and the actual failure occurs at the different scattered points I have shown the scattered points a border line divides the safe region from unsafe region from various combination of sigma m and sigma i it is indicated that various combinations of sigma m and sigma i sigma m may be dominating sigma a may be dominating according to that even the point will be obtained and I need to divide it from the safe region and unsafe region I should see that the component must be designed under the safe region these lines are the important lines I need to consider for safe design of the component at Soderbergh line is more conservative the failure criteria and there is no need to consider even yielding in this case because it is at the bottom most line if you design for the Soderbergh line if you consider that line for the design purpose then there is no need there is no further need to think whether the component will yield because it is well below the yield line there will not be the failure due to yielding the yield line is constructed connecting syd on both axes it is called the limit on the first cycle of stress this is because if a part yields it has failed because if we say that the point falls in the region of safety of the fatigue however if the yielding happens the component is already failed we cannot design it for fatigue in that case therefore Soderbergh line then this yield line these are very important then the Gerber line this parabola fits the failure points of the test data in the best possible way Goodman line fits beneath the scatter of this data both the Gerber parabola and the Goodman line intersect at s e on the ordinate to the s ut on the abscissa the Goodman line is more safe from the design consideration because it is completely inside the Gerber parabola and inside the failure points the equation of state line I can apply x upon a plus y upon b equals 1 whether where this x and y are the axes and a and b are the respective intercepts I can use the Soderbergh line equation as sigma m by syt plus sigma a by sc equals 1 and for the Goodman line the equation is sm upon sigma m upon s ut plus sigma a upon sc equals 1 the Goodman line is widely used as a criteria of fatigue failure because it is safe the equation of state line is simple it is not necessary to construct a scale diagram and rough sketch is enough to construct the fatigue diagram thank you