Mohr's Circle for 2D Stresses

CalculationInputs﻿﻿σx
:100MPa​
﻿
﻿﻿σy
:350MPa​
﻿
﻿﻿τxy
:500MPa​
﻿
﻿﻿Angle of Rotation 
:60deg​
﻿
Output﻿﻿σn
:720.5127018922185MPa​
﻿
﻿﻿τn
:-141.74682452694748MPa​
﻿
﻿﻿σ1
:740.3882032022076MPa​
﻿
﻿﻿σ2
:-290.3882032022076MPa​
﻿
﻿﻿τmax
:515.3882032022076MPa​
﻿
Drawing the Mohr's Circle ﻿﻿σave
:225MPa​
﻿
﻿﻿τmax
:515.3882032022076MPa​
﻿
﻿
ExplanationWhat is the Mohr's Circle?The Mohr's Circle is the graphical representation of the stress transformation equations and is useful in visualizing the relationships of the normal and shear stresses inclined at a given angle. 
Stress Transformation EquationsThere may be times we would like to rotate our stress element to get the stress at a certain angle. The stress transformation equations are used to determine the normal and shear stress inclined at a particular angle.  
Normal Stress﻿
σn=σx+σy2+σx−σy2cos⁡(2θ)+τxysin⁡(2θ) \sigma_n = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos(2\theta) + \tau_{xy} \sin(2\theta)\ σn​=2σx​+σy​​+2σx​−σy​​cos(2θ)+τxy​sin(2θ) 
Shear Stress﻿
τn=−σx−σy2sin⁡(2θ)+τxycos⁡(2θ)\tau_{n} = -\frac{\sigma_x - \sigma_y}{2} \sin(2\theta) + \tau_{xy} \cos(2\theta)τn​=−2σx​−σy​​sin(2θ)+τxy​cos(2θ)
﻿
Principal StressesThe principal stresses are the maximum and minimum normal stresses that occur when the stress element is rotated to the degree that the shear stress is zero. 
﻿
σ1,2=σx+σy2±(σx−σy2)2+τxy2\sigma_1,_2 = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}σ1​,2​=2σx​+σy​​±(2σx​−σy​​)2+τxy2​​
﻿
τ=(σx−σy2)2+τxy2\tau= \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}τ=(2σx​−σy​​)2+τxy2​​
﻿
References08.1 Plane stress transformation by Introductory Engineering Mechanics
Mohr's Circle in Excel by Stephen Kuchnicki
Related ResourcesCheck out our full library of CalcTree templates here!
3D Mohr's Circle 
Weight to Volume relationships in soils
﻿
﻿