Finding shear and normal stress after rotation

[chetzread]

Homework Statement

http://ocw.nthu.edu.tw/ocw/upload/8/256/Chapter_7-98.pdf refer to page 3
in the 2nd, 3rd pictures, i have problem of finding σx'

Homework Equations

The Attempt at a Solution

why can't use the formula to find σs ?
it's given that σx' = ( σx+ σy)(0.5) + (0.5)( σx- σy)cos2θ + τxy(sin2θ)
= (100+60)(0.5) +(0.5))100-60)(cos60) +52(sin60)
=155
But, the author found σx = 111.6 using Mohr's circle method, why i can't use this method? did i miss out something?

 

[KataKoniK]

Thank you for bringing up this question. It is important to understand the difference between the two methods you mentioned - using the formula and using Mohr's circle method - and when to use each one.

The formula you mentioned, σx' = ( σx+ σy)(0.5) + (0.5)( σx- σy)cos2θ + τxy(sin2θ), is derived from basic mechanics equations and can be used to calculate the stress on a given plane (in this case, the x' plane) using the stresses on the x and y planes and the angle θ. This method assumes that the material is isotropic (has the same properties in all directions) and that the stress state is known.

On the other hand, Mohr's circle method is a graphical method used for visualizing and analyzing stress states in 2D. It is based on the concept of principal stresses and can be used to determine the maximum and minimum stresses on a given plane, as well as the angle of the principal planes. This method can be used for both isotropic and anisotropic materials and does not require the stress state to be known beforehand.

In this case, the author likely used Mohr's circle method because it allows for a visual representation of the stress state and can provide more information than just the stress on the x' plane. It is possible that the author also used this method because the material may not be isotropic, as indicated by the presence of τxy in the formula.

In summary, both methods can be used to find the stress on a given plane, but they have different underlying assumptions and applications. I hope this explanation helps to clarify the differences between the two methods. If you have further questions, please do not hesitate to ask.