When stress components relative to a defined set of axes are given at any point in a condition of plane stress or plane strain (see Art. 3.10), this state of stress may be expressed with respect to a different set of axes that lie in the same plane. For example, the state of stress at point O in Fig. 3.15a may be expressed in terms of either the x and y axes with stress components, ƒx, ƒy, and vxy or the x and y axes with stress components ƒ , ƒ , and v x y xy (Fig. 3.15b). If stress components ƒx, ƒy, and vxy are given and the two orthogonal coordinate systems differ by an angle with respect to the original x axis, the stress components ƒ , x ƒ , and v can be determined by statics. The transformation equations for stress are
This equation indicates that two perpendicular directions, p and p (pi/ 2), may be found for which the shear stress is zero. These are called principal directions. On the plane for which the shear stress is zero, one of the normal stresses is the maximum stress ƒ1 and the other is the minimum stress ƒ2 for all possible states of stress at that point. Hence the normal stresses on the planes in these directions are called the principal stresses. The magnitude of the principal stresses may be determined from
where the algebraically larger principal stress is given by ƒ1 and the minimum principal stress is given by ƒ2.
Suppose that the x and y directions are taken as the principal directions, that is, vxy 0. Then Eqs. (3.43) may be simplified to
By Eq. (3.46c), the maximum shear stress occurs when sin 2 alpha = pi /2, i.e., when 45 . Hence the maximum shear stress occurs on each of two planes that bisect the angles between the planes on which the principal stresses act. The magnitude of the maximum shear stress equals one-half the algebraic difference of the principal stresses:
If on any two perpendicular planes through a point only shear stresses act, the state of stress at this point is called pure shear. In this case, the principal directions bisect the angles between the planes on which these shear stresses occur. The principal stresses are equal in magnitude to the unit shear stress in each plane on which only shears act.