Principal Stresses

The states of stress illustrated in diagrams above are termed complex in the sense that both normal and shearing stresses are present at any point. This has been brought out in preceding section that the state of stress at the similar point may change if the axes of coordinates are changed. This is possible to find planes passing through any point such they will not carry any shearing stress. Planes that carry only normal stress and no shearing stress are termed principal planes and stresses acting upon them are known as principal stresses. This can be seen that in plane stress state in above diagrams, that there will be two principal planes normal to which will be inclined at angle q_p, anticlockwise with x-axis such,

                                                    tan 2Θ_p  = (2Τ_xy/(σ_x - σ_y))...........................8

Eqn8 represents a set of planes mutually orthogonal. One principal stress is associated along with each of the principal planes - one being minimum and other maximum If or, and on signify maximum and minimum principal stresses respectively, is that:

                                        σ_P1 = (σ_X +σ _Y)/2 + √{(((σ_X - σ_Y)/2)²) + t²_XY} .......................9

                                        σ_P2 = (σ_X + σ_Y)/2 - √{(((σ_X -σ_Y)/2)²) + t²_XY}.......................10

There will be another set of planes passing throughout a point in the body, which will carry maximum shearing stress. This can be shown that this set of planes would be inclined to principal planes at angle of 45⁰. This was shown in previous section with reference to diagram that the plane, on which t_max = s/2 was acting, was at q = 45⁰ with axis of the part A₂. This was the maximum shearing stress as referred to now. The angle of inclination of normal to planes of maximum shearing stress, denoted by q_t, is described by:

                                                  tanθ_t = -((σ_X - σ_Y)/2Τ_XY)............................. 11

The maximum sharing stress is:

                                             Τ_max = + √{(((σ_X - σ_Y)/2Τ_XY)²) +Τ²_XY}

                                                       = (σ_P1 - σ_P2)/2..............................................12

This may be noted that the maximum shearing stress is half of the difference of maximum and minimum principal stresses. This will be seen in previous diagram that one of the principal P planes is the transverse section and thus the maximum principal stress is s =P/A while A minimum principal stress is zero, whereas:

                                                                         Τ_max = σ/2.................7

Generally when stresses act in three dimensions there are three orthogonal principal planes passing throughout any point. Hence there will be three principal stresses:

• One maximum,
• One minimum and
• The third among the two

The maximum shearing stress will then be the half of the difference of minimum and maximum principal stresses. A very significant relationship among the normal stresses on any two sets of orthogonal planes may be derived from eqn. 9 and 10. By addition of:

                                                  σ_P1 + σ_P2 =σ_X + σ_{Y......................................................}13

This means that sum of normal stresses on any one set of orthogonal planes passing by a point in a body remains const. This total is termed as stress in variant.