Transformation of Plane Stress

Consider a point on a cube experiencing plane stress, defined by its stress components.

If the element rotates by an angle, its stress components change. This change is determined by considering a prismatic element with faces perpendicular to its axes.

By considering the area of the oblique face, the areas of the vertical and horizontal faces can be calculated using trigonometric functions of the rotation angle.

The forces exerted on these faces are represented, and it is assumed that no forces act on the triangular faces of the element. The equilibrium equations are then written using components along the rotated axes.

Upon solving these equations, the normal and shearing stresses are derived. The normal and shearing stresses are then re-expressed using trigonometric relations.

An expression for the normal stress on the rotated vertical axis is obtained by substituting the rotation angle in one of the previous expressions with a new angle.

It is observed that the sum of the normal stresses exerted on a cubic element remains independent of the element's orientation.