PLANE STRAIN AND TRANSFORMATIONS

Plane strain transformations tool was developed to calculate normal strains and shear strain  at a specific point for plane strain state (εzzxzy=0) after the element is rotated by θ around the Z-axis.  If the plane strains are known for a specific point on a member, then plane strains 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.

Plane Strain Transformation
INPUT PARAMETERS
Parameter Symbol Value Unit
Normal strain
εx μm/m (μin/in)
Normal strain εy
Shear strain γxy
Transformation angle θ deg

 



Note: Use dot "." as decimal separator.


RESULTS
Parameter Symbol Value Unit
Normal strain
after transformation
εx' --- μm/m (μin/in)
Normal strain
after transformation
εy' ---
Shear strain
after transformation
γxy' ---



Definitions:

Normal Strain: The ratio of length change to original length of the material. ε=σ/E

Shear Strain: The angular distortion on element caused by shear stress. γ=τ/G. 

Plane Strain: A state where normal and shear strains occur within a plane and no strains occur perpendicular to this plane. (εz= γxz = γyz =0). This situation occurs in a plate subjected along its edges to uniformly distributed loads and restrained from expanding or contracting laterally by smooth, rigid and fixed supports. An example to this can be rolling of the sheet metal between rollers. In this situation, expansion of the metal is constrained by rollers in perpendicular direction.

Plane strain example - Sheet metal between rollers
List of Equations:

Parameter Symbol Formula
Normal strain
after transformation
εx' x+εy)/2+Cos(2θ)(εxy)/2+γxySin(2θ)/2
Normal strain
after transformation
εy' (εx+εy)/2-Cos(2θ)(εxy)/2-γxySin(2θ)/2
Shear strain
after transformation
γxy' -Sin(2θ)(εxy)+γxyCos(2θ)

Reference: