PRINCIPAL & MAXIMUM SHEAR STRESS CALCULATOR FOR PLANE STRESS

Principal stress calculator is developed to calculate principal and maximum shear stresses, principal and maximum shear stress angles and Von Mises stress at a specific point for plane stress (σzzxzy=0). Mohr's Circle is also drawn according to input parameters.

The formulas used for the calculations are given in the List of Equations section.

Calculator:

Principal stresses and maximum shearing stress calculation
INPUT PARAMETERS
Parameter Symbol Value Unit
Normal stress σx
Normal stress σy
Shear stress
τxy

 



Note: Use dot "." as decimal separator.


RESULTS
Parameter Symbol Value Unit
Maximum principal stress σmax --- MPa
Minimum principal stress σmin ---
Maximum shear stress*  τmax ---
Average principal stress σavg ---
Von Misses stress σmises ---
Angle of principal stresses ** θp --- deg
Angle of maximum shear stress ** θs ---


Note 1: *Maximum shear stress given in results is the maximum in-plane shear stress. For out-plane shear stress, check 3D stress analysis.

Note 2:** There are 2 values for this parameter. The first value is shown in the table and the other value is 90° apart from the first one.

Definitions:

Maximum Shear Stress Angle: The angle of orientation at which maximum in-plane shear stress occurs for a specific point.

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.

Principal Angle: The angle of orientation at which principal stresses occur for a specific point.

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.

Supplements:

Link Usage
Yield Criteria for Ductile Material After calculation of principal stresses, yield criteria can be checked for ductile material.

List of Equations:

Step Parameter/Condition Symbol Equation
1 Maximum principal stress σmax Equation for maximum principal stress in plane stress situation
2 Minimum principal stress σmin Equation for minimum principal stress in plane stress situation
3 Principal angle θp Equation for principal angle in plane stress situation
4 Maximum shear stress angle θs Equation for maximum shear stress angle in plane stress situation
5 Maximum shear stress τmax Equation for maximum shear stress in plane stress situation

Symbol Parameter
σx Normal stress in X direction
σy Normal stress in Y direction
τxy Shear stress perpendicular to X axis and in Y direction.

Examples:

Link Usage
Torsion Of Solid Shaft An example about the calculation of torsional stress on stepped shaft. After calculation of torsional stress,  principal stresses are calculated and evaluation of yield criteria of material is done with these stresses.

Reference:
  • Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGraw-Hill