Mohr's Circle Formula: Understanding Why σ_xl Disappears

[foo9008]

Homework Statement

https://web.iit.edu/sites/web/files...Academic Resource Center/pdfs/Mohr_Circle.pdf[/B]
why will σ_xl disappear in the equation ? [ [σ_xl - (σx -σy)/ 2 ] ^2 ] = ( [ (σx -σy)/ 2] ^2 ) is it wrong ?

Homework Equations

The Attempt at a Solution

 

[haruspex]

Homework Statement

https://web.iit.edu/sites/web/files/departments/academic-affairs/Academic Resource Center/pdfs/Mohr_Circle.pdf[/B]
why will σ_xl disappear in the equation ? [ [σ_xl - (σx -σy)/ 2 ] ^2 ] = ( [ (σx -σy)/ 2] ^2 ) is it wrong ?

Homework Equations

The Attempt at a Solution

I don't think the above equation is supposed to follow from a simple manipulation of one side to produce the other. Rather, this equation is the result of some sequence of steps. Note that you have misquoted it: on the left hand side it should be σ_x+σ_y, not minus.
See if https://en.m.wikipedia.org/wiki/Mohr's_circle helps.

 

[foo9008]

I don't think the above equation is supposed to follow from a simple manipulation of one side to produce the other. Rather, this equation is the result of some sequence of steps. Note that you have misquoted it: on the left hand side it should be σ_x+σ_y, not minus.
See if https://en.m.wikipedia.org/wiki/Mohr's_circle helps.

can you show me the proof ? the link doesn't provide the proof

 

[haruspex]

can you show me the proof ? the link doesn't provide the proof

It has what looks to me as a claimed proof. At "Equation of the Mohr Circle" it obtains expressions for σ_n and τ_n as functions of θ. Then it eliminates θ between these two equations to obtain a relationship between σ_n and τ_n. This produces the equation you queried.