Transformation of Plane Strain

Consider a long bar subjected to uniformly distributed loads on its sides.

In such bars, the transformation of plane strain under a rotation of coordinate axes involves states of plane strain where deformations occur within parallel planes.

A plane strain state at point O has strain components associated with the x and y axes. The strain components are then expressed in terms of the angle θ.

An expression is derived for normal strain along a line forming an arbitrary angle with the x-axis.

Then, the normal strain in the direction of the bisector of the angle formed by the x and y axes is determined. The shearing strain is then expressed in terms of these normal strains.

Considering trigonometric relations, the equations for plane strain transformation under axis rotation are derived by calculating normal strain along the bisector of the x' and y' axes, and expressing shearing strain in terms of normal strain.