# Goodman’s Criteria – Theories of Elastic Failure – Strength of Materials

Hello friends here in this video we will see what is meant by Goodman’s criteria along with the Goodman’s line now here we have a diagram which is also called as the diagram of Goodman or Goodman’s criteria your Goodman’s criteria is right down Goodman’s criteria is used when ultimate stress is considered while designing so when we are taking the ultimate stress into account then we have to use the Goodman’s criteria now Goodman’s criteria basically it consists of it is a graph of variable stresses on y axis variable stresses versus mean stress so it is a graph of variable stresses versus the mean stress now here as we see the line which is joining point capital a to capital B it is called as the Goodman line or also called as failure stress line at Point a we have Sigma e which is called as endurance stress at point B we have Sigma you called as ultimate stress joining them will give us the Goodman line because ultimate stress is the maximum value so this line won’t cross Sigma u it will end only on Sigma u similarly it will end on Sigma e next what we are doing now after this dividing this Sigma E by using factor of safety so Sigma E upon factor of safety that will reduce the stress because factor of safety is always greater than 1 so dividing Sigma E by FS we are getting a stress value which is at Point C similarly Sigma u value is greater dividing Sigma u by factor of safety we get a point D now after getting point C and D we will move parallel to a B and then join a line C to D which is parallel to the Goodman line next after this we will take the midpoint of this line CD point P which is called as a design point and on x-axis we would be getting mean stress and on Y we would be getting variable stress now once we have understood this next I can write down from this the first thing is from the similarity of triangles see OD + PQ d from the similarities of triangles see OD this triangle and PQ d the other triangle we have P Q upon Co that is the base of the triangle PQ upon Co is equal to QD upon OD that is the side of smaller triangle upon the side of the bigger triangle QD upon OD next QD it can be written as it is QD can be written as from OD we can subtract OQ that will give us QD so OD – OQ upon OD will remain as it is so therefore P Q upon Co is equal to OD upon OD – OQ upon OD so from this OD and OD gets cancelled out we have one here so next we have PQ upon C o is equal to one minus o Q upon OD now after this we can say that P Q is nothing but variable stress Sigma V so instead of PQ I can write down Sigma V variable stress see Oh as we can see Co is Sigma II upon factor of safety Sigma II that is the indian stress upon factor of safety is equal to one minus now oq o Q is equal to Sigma M + OD OD is equal to Sigma u upon FS that is factor of safety so Sigma u upon factor of safety now once we have reached at this stage I will keep Sigma V on one side that is therefore the variable stress so now your I keep Sigma V variable stress on one side so therefore variable stress will be on one side and I will shift this term Sigma upon FIS onto the other side so Sigma e upon fo s and here previously we have 1 minus Sigma M upon Sigma u divided by F voice now I will simplify this so we have Sigma B is equal to Sigma Y upon F is I will give it as it is next this fos will get multiplied above so here this will be minus 1 minus Sigma M into F s upon Sigma u then this will be Sigma u minus Sigma M fo s upon Sigma u therefore Sigma e upon Sigma u upon Sigma u we can keep it one above the other – Sigma M fos upon Sigma u next Sigma u will get cancelled out so we have one which remains here 1 minus Sigma M into F is upon Sigma u now I will take F is common or from the bracket so we have Sigma V is equal to Sigma I upon fo s into F is I take it common so inside the bracket we have 1 upon F is minus Sigma M upon Sigma u so f is f word gets cancelled so what remains here is therefore Sigma V is equal to Sigma E into 1 upon fo s minus Sigma M upon Sigma u now this equation it can be simplified so Sigma V upon Sigma E is equal to 1 upon F is minus Sigma M upon Sigma u therefore keeping 1 upon fo s on one side bringing these terms on to the other side so this term Sigma M by Sigma u which is negative will become positive now this is the equation of Goodman’s criteria neglecting stress concentration now if in this the stress concentration is considered then the equation I can write it directly it will modify to therefore 1 upon fos is equal to Sigma M upon Sigma u plus Sigma V into K suffix F divided by Sigma e into surface into K sighs so this is another equation of goodman’s considering the stress concentration factor so this is equation of Goodman’s criteria considering stress concentration here I will explain the terms where K suffix F it is called as fatigue factor and this is for stress concentration next we have K surface it is called as the surface finish factor and at last we have K suffix s Z which is called as size factor so these factors are to be considered here we have KF as the fatigue factor for stress concentration k su r which is the surface factor KS z which is the size factor these factors are are to be considered while designing any component and previously the equation which I have written it was neglecting all these factors so in this video we have seen the Goodman’s criteria along with the diagram Goodman’s line and then the expression