# MOHR'S CIRCLE FOR 3D STRESS ANALYSIS

Mohr's circle for 3d stress analysis calculator was developed to calculate 3d principal stresses, maximum shear stresses, and Von Mises stress at a specific point for spatial stresses. 3d Mohr's Circle 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.

Principal and maximum shear stress formulas used for the calculations are given in the "List of Equations" section.

### 3D Mohr's Circle Calculator: INPUT PARAMETERS Parameter Value Unit Normal stress (σx) MPa psi Normal stress (σy) Normal stress (σz) Shear stress (τxy) Shear stress (τyz) Shear stress (τxz)

Note: Use dot "." as decimal separator.

 RESULTS Parameter 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

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 Formula Characteristic polynomial equation σ3-Aσ2+Bσ-C=0 Polynomial coefficient (A) =σx+σy+σz =σ'1+σ'2+σ'3 Polynomial coefficient (B) =σxσy+σyσz+σxσz-(τxy)2-(τyz)2-(τxz)2 =σ'1σ'2+σ'2σ'3+σ'1σ'3 Polynomial coefficient (C) =σxσyσz+2τxyτyzτxz-σx(τyz)2-σy(τxz)2-σz(τxy)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) (σ2-σ3)/2 Max shear stress -2 (τmax2) (σ1-σ3)/2 Max shear stress -3 (τmax3) (σ1-σ2)/2

### Reference:

• Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGraw-Hill
• Budynas.R , Nisbett.K . (2008) . Shigley's Mechanical Engineering Design . 8th edition.  McGraw-Hill