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 σx+σy, not minus.foo9008 said:
can you show me the proof ? the link doesn't provide the proofharuspex said:
It has what looks to me as a claimed proof. At "Equation of the Mohr Circle" it obtains expressions for σn and τn as functions of θ. Then it eliminates θ between these two equations to obtain a relationship between σn and τn. This produces the equation you queried.foo9008 said:
Mohr's Circle Formula is a graphical method used to determine the principal stresses and maximum shear stress at a point on a two-dimensional stress element.
Mohr's Circle is constructed by plotting the normal stress (σ) on the horizontal axis and the shear stress (τ) on the vertical axis. The center of the circle represents the average normal stress, and the radius of the circle represents the maximum shear stress.
The σ_xl term disappears because it is a normal stress acting in the x-direction, which is the axis of rotation for the stress element. This means that it has no effect on the shear stress and can be disregarded in the calculations.
Mohr's Circle Formula is used in engineering to analyze and design structures subject to stress. It allows engineers to determine the maximum stresses and their orientation at a point, which is essential in ensuring the safety and stability of structures.
Mohr's Circle Formula is limited to two-dimensional stress analysis and cannot be used for three-dimensional problems. It also assumes that the material is homogeneous and isotropic, and the stress state remains linearly elastic.
