# PLANE STRAIN AND PRINCIPAL STRAINS

Principal strains calculation tool was developed to calculate principal strains and maximum in-plane shear strain at a specific point for plane strain state (εzzxzy=0) .

The formulas used for the calculations are given in the "List of Equations" section. INPUT PARAMETERS Parameter Value Unit Normal strain ( εx) μm/m (μin/in) Normal strain (εy) Shear strain (γxy)

Note: Use dot "." as decimal separator.

 RESULTS Parameter Value Unit Maximum normal strain (εmax) --- μm/m (μin/in) Minimum normal strain (εmin) --- Maximum shear strain (in-plane) (γmax (in-plane)) --- Angle of principal strain (θp) --- deg

### Definitions:

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

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

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

Principal Strain: Maximum and minimum normal strain possible for a specific point on a structural element. Shear strain is 0 at the orientation where principal strain occurs.

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

### List of Equations:

 Parameter Formula Maximum normal strain (εmax) (εx+εy)/2+(((εx-εy)/2)2+(γxy/2)2)0.5 Minimum normal strain (εmin) (εx+εy)/2-(((εx-εy)/2)2+(γxy/2)2)0.5 Maximum shear strain (in-plane) ( γmax (in-plane)) ((εx-εy)2+(γxy)2)0.5 Principal angle (θp) [atan(γxy/(εx-εy))]/2

### Reference:

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