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Mohr's Circle

General Case

Mohr's circle represents the stress-transformation equation graphically and shows how the normal and shear stress components vary as the plane on which they act is oriented in different directions.

The red color's state of stress on the right corresponding to the red point on the circumference on the left.

[s-(s_x+s_y)/2]²+t²=[(s_x-s_y)/2]²+t_xy²

This equation is of the form

(s-a)²+(t-b)²=r²

where a=(s_x+s_y)/2,    b=0,     r²=[(s_x-s_y)/2]²+t_xy²

It is evident that the radius r is indeed the maximum shear stress in the x-y plane.

The maximum and minimum normal stresses occur along the  s axis

s_1,2= a ± r =(s_x+s_y)/2 ± {[(s_x-s_y)/2]²+t_xy²}^1/2

A rotation of of the radium by an angle of 2q on the Mohr 's circle gives the state of stress for a coordinate of q in the same direction in the material, this is animated by the state of plane stress for a rotated coordinate system.

Notice:

• when the diameter becomes horizontal at the right, maximum normal stresses are obtained.
• when the diameter becomes vertical, the state of stress obtained contains the principal  shear stress.
• when the diameter becomes horizontal at the left, minimum normal stresses are obtained.

When  a = (s_x+s_y)/2=0,

Special Case

This is a special case for Mohr's circle, The coordinate origin is located in the center of  Mohr's circle. The normal stress changes from tension (positive) to compression (negative).

The red color's state of stress on the right corresponding to the red point on the circumference on the left.

s²+t²=r²

where  r=t_xy

The radius r equals the maximum shear stress in the x-y plane.

The maximum and minimum normal stresses occur along the  s axis

s_1,2=± r =±t_xy

A rotation of of the radium by an angle of 2q on the Mohr 's circle gives the state of stress for a coordinate of q in the same direction in the material, this is animated by the state of plane stress for a rotated coordinate system.

Notice:

• when the diameter becomes horizontal at the right, maximum normal stresses are obtained.
• when the diameter becomes vertical, the state of stress obtained contains the principal  shear stress.
• when the diameter becomes horizontal at the left, minimum normal stresses are obtained.

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