3D STRESS ANALYSIS WITH MOHR'S CIRCLE


3D stress analysis calculation tool was 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 out-plane 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:

3D stress analysis with mohr's circle
INPUT PARAMETERS
Parameter Symbol Value Unit
Normal stress σx
Normal stress σy
Normal stress σz
Shear stress τxy
Shear stress τyz
Shear stress τxz

 



Note: Use dot "." as decimal separator.


RESULTS
Parameter Symbol Value Unit
Principal stress-1 σ1 --- MPa
Principal stress-2 σ2 ---
Principal stress-3 σ3 ---
Max shear stress -1 τmax1 ---
Max shear stress -2 τmax2 ---
Max shear stress -3 τmax3 ---
Von Mises stress σv ---

    



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
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 - σ3-Aσ2+Bσ-C=0
Polynomial coefficient A xyz
=σ'1+σ'2+σ'3
Polynomial coefficient B xσyyσzxσz-(τxy)2-(τyz)2-(τxz)2
=σ'1σ'2+σ'2σ'3+σ'1σ'3
Polynomial coefficient C xσyσz+2τxyτyzτxzxyz)2yxz)2zxy)2
=σ'1σ'2σ'3
Principal stress-1 σ1 max(σ'1,σ'2,σ'3)
Principal stress-2 σ2 A-σ'1-σ'2
Principal stress-3 σ3 min(σ'1,σ'2,σ'3)
Max shear stress -1 τmax1 23)/2
Max shear stress -2 τmax2 13)/2
Max shear stress -3 τmax3 12)/2

Reference: