PLANE STRESS AND TRANSFORMATIONS

Plane stress transformation tool was developed to calculate normal stresses and shear stress at a specific point for plane stress state (σzzxzy=0) after the element is rotated by θ around the Z-axis. Results are also shown with Mohr's Circle representation. If the plane stresses are known for a member, then plane stresses for different orientation (in the same plane) can be calculated with this calculator.

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



Calculator:

Plane stress transformation
INPUT PARAMETERS
Parameter Symbol Value Unit
Normal stress
σx
Normal stress
σy
Shear stress
τxy
Transformation angle θ deg

 



Note: Use dot "." as decimal separator.


RESULTS
Parameter Symbol Value Unit
Normal stress
after transformation
σx' --- MPa
Normal stress
after transformation
σy' ---
Shear stress
after transformation
τxy' ---



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

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
Normal stress
after transformation
σx' x+σy)/2+Cos(2θ)(σxy)/2+τxySin(2θ)
Normal stress
after transformation
σy' x+σy)/2-Cos(2θ)(σxy)/2-τxySin(2θ)
Shear stress
after transformation
τxy' -Sin(2θ)(σxy)/2+τxyCos(2θ)

Examples:

Link Usage
Pressure Vessel An example about the calculation of stresses on a pressure vessel, evaluation of yield criteria of material and stress transformation to find shear and perpendicular stresses on welding of the cylindrical body of the pressure vessel.

Reference: