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 (ε_{z}=γ_{zx}=γ_{zy}=0)
after the element is rotated by θ around the Zaxis. 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.
Note: Use dot "." as decimal separator.
RESULTS 
Parameter 
Symbol 
Value 
Unit 
Normal strain
after transformation 
ε_{x}' 


μm/m (μin/in) 
Normal strain
after transformation 
ε_{y}' 


Shear strain after transformation 
γ_{xy}' 


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. (ε_{z}= γ_{xz} = γ_{yz} =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 
ε_{x}' 
(ε_{x}+ε_{y})/2+Cos(2θ)(ε_{x}ε_{y})/2+γ_{xy}Sin(2θ)/2 
Normal strain
after transformation 
ε_{y}' 
(ε_{x}+ε_{y})/2Cos(2θ)(ε_{x}ε_{y})/2γ_{xy}Sin(2θ)/2 
Shear strain after transformation 
γ_{xy}' 
Sin(2θ)(ε_{x}ε_{y})+γ_{xy}Cos(2θ) 
Reference: