3D STRESS ANALYSIS WITH MOHR'S CIRCLE
3D stress analysis calculation tool is developed to calculate principal stresses, maximum
shear stresses, and Von Mises stress at a specific point for spatial stresses . Calculator can be used to calculate
outplane shear stress for plane stress situation. Mohr's Circle for 3D stress analysis is also
drawn according to input parameters.
The formulas used for the calculations are given in the List of Equations
section.
Calculator:
Note: Use dot "." as decimal separator.
RESULTS

Parameter 
Symbol 
Value 
Unit 
Principal stress1 
σ_{1} 


MPa

Principal stress2 
σ_{2} 


Principal stress3 
σ_{3} 


Max shear stress 1 
τ_{max1} 


Max shear stress 2 
τ_{max2} 


Max shear stress 3 
τ_{max3} 


Von Mises stress 
σ_{v} 


Definitions:
Mohr’s Circle: A graphical method to represent the plane stress (also strain) relations.
It’s a very effective way to visualize a specific point’s stress states, stress transformations for an angle,
principal and maximum shear stresses.
Normal Stress: Stress acts perpendicular to the surface (cross section).
Plane Stress: A loading situation on a cubic element where two faces the element is free of any stress.
Such a situation occurs on free surface of a structural element or machine component, at any point of the
surface of that element which is not subjected to an external force. Another example for plane stress is structures which are built from sheet metals where stresses across the thickness are negligible.

Plane stress example  Free surface of structural element 
Principal Stress: Maximum and minimum normal stress possible
for a specific point on a structural element. Shear stress is 0 at the orientation where principal stresses occur.
Shear stress: A form of a stress acts parallel to the surface (cross section) which has a cutting nature.
Stress: Average force per unit area which results strain of material.
List of Equations:
Parameter 
Symbol 
Formula 
Characteristic polynomial equation 
 
σ^{3}Aσ^{2}+BσC=0 
Polynomial coefficient 
A 
=σ_{x}+σ_{y}+σ_{z}
=σ'_{1}+σ'_{2}+σ'_{3} 
Polynomial coefficient 
B 
=σ_{x}σ_{y}+σ_{y}σ_{z}+σ_{x}σ_{z}(τ_{xy})^{2}(τ_{yz})^{2}(τ_{xz})^{2}
=σ'_{1}σ'_{2}+σ'_{2}σ'_{3}+σ'_{1}σ'_{3} 
Polynomial coefficient 
C 
=σ_{x}σ_{y}σ_{z}+2τ_{xy}τ_{yz}τ_{xz}σ_{x}(τ_{yz})^{2}σ_{y}(τ_{xz})^{2}σ_{z}(τ_{xy})^{2}
=σ'_{1}σ'_{2}σ'_{3} 
Principal stress1 
σ_{1} 
max(σ'_{1},σ'_{2},σ'_{3}) 
Principal stress2 
σ_{2} 
Aσ'_{1}σ'_{2} 
Principal stress3 
σ_{3} 
min(σ'_{1},σ'_{2},σ'_{3}) 
Max shear stress 1 
τ_{max1} 
(σ_{2}σ_{3})/2 
Max shear stress 2 
τ_{max2} 
(σ_{1}σ_{3})/2 
Max shear stress 3 
τ_{max3} 
(σ_{1}σ_{2})/2 
Reference:
 Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGrawHill
 Budynas.R , Nisbett.K . (2008) . Shigley's Mechanical Engineering Design . 8th edition. McGrawHill