General State of Stress in Two Dimensions:

Normal component on the plane = P cos θ

                                                     = σ_x × A cos θ

Tangential component = - P sin θ  = - σ_x × A sin θ

Normal stress on the plane   = ( σ _x × A cos θ)/(A/ cos θ) = σ_x × cos2θ

Shear stress on the plane         =  - σ _x × Asin θ /(A /cos θ)

                                                 = - σ_x × cos θ sin θ

                  Figure

If on the cross sectional area of the original solid shear stress τ_xy is applied, its components on the inclined plane might be evaluated as,

Normal stress component = τ_xy cos θ sin θ

Shear stress component           = τ_xy cos2θ

By a same analysis (your exercise), you might verify the following:

If a normal stress of σ_y is applied on the solid, then the stress components on a plane whose normal is inclined at θ to the x axis are specified by

Normal stress = σ_y sin2θ

 Shear stress    = σ_y sin θ cos θ

If a shear stress of τ_yx is applied on the solid, the stress components on the inclined plane are specified by (with τ_yx = τ_xy).

Normal stress = τ_xy sin θ cos θ

 Shear stress    = τ_xy sin2θ

As a general state of stress in two dimensions is defined by the stress components σ_x, σ_y and τ_xy, as shown in Figure, general expressions for normal and shear stress components might be obtained by algebraic sum of the respective components from Eqs. (22) to (29), as given below :

σ_n = σ_x cos2 θ + σ_y sin2 θ +2 τ_xy cos θ sin θ

τ_nt = - σ_x cos θ sin θ + σ_y cos θ sin θ + τ_xy (cos2 θ - sin2 θ)

On further simplification, we obtain

σ_n = (σ _x + σ _y) /2 +(( σ _x - σ _y)/2 )cos 2θ + τ_xy sin 2θ

τ_nt =(( σ _y - σ_x) /2) sin 2θ + τ_xy cos 2θ

Figure: General State of Stress in Two Dimensions

Application of Eqs. and shall be elaborately dealt with in this unit. However, the significance of the equations must be stressed at this stage. When we are carrying out stress analysis on solids, we might be evaluating stress components on a set of mutually perpendicular planes, such like σ_x, σ_y & τ_xy. The magnitudes of these components might not always be adequate to decide whether the solid is secure or not, and we needs evaluation of stress components on other particular planes, or determination of planes on which the stress components have extreme value.