23.7:
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.
23.7:
Transformation of Plane Strain
When analyzing elongated structures like bars subjected to uniformly distributed loads, it is essential to understand the transformation of plane strain when coordinate axes are rotated. This transformation helps to assess how material deformation characteristics vary with orientation, which is crucial in materials science and structural engineering.
Under plane strain conditions, typical for members where one dimension significantly exceeds the others, deformations and resultant strains are notable only in a single plane. At any point, say O, the important strain components are those along the x and y axes, and the strain components along the z-axis are negligible. These components comprehensively describe the deformation state within the xy-plane.
A significant case is the calculation of normal strain along the bisectors of the angle between the x and y axes, which is 45 degrees. In this case, considering this angle simplifies the expression, reflecting how the combination of normal strains and shearing strain influences deformation along this line.
When the coordinate system is rotated to align with the bisector of the x and y axes, the strain components in the new coordinates can be recalculated using trigonometric relations derived from elasticity theory. Specifically, the normal strains in the directions forming an arbitrary angle with the original axes and the corresponding shearing strains can be determined in terms of the original strain components.
