Normal and Shear Stresses - Civil Engineering (CE) PDF Download

Normal and shear stresses 
Let the traction vector t_(n) for a given current position x at time t act on an arbitrarily oriented surface element characterized by an outward unit normal vector n.
The traction vector t_(n) may be resolved into the sum of a vector along the normal n to the plane, denoted by  and a vector perpendicular to n denoted by  From the results in section 2.3.3, it could be seen that



where


It could easily be verified that n·m = 0. This means m is a vector embedded in the surface. Then, σ_n is called the normal traction and τ_n is called the shear traction. As the names suggest, normal traction acts perpendicular to the surface and shear traction acts tangential or parallel to the surface.
Since,  we obtain the useful relation



where we have made use of the property that n · m = 0. 
It could be seen from figure 4.3 that the stress components corresponding to Cartesian basis, σ_xx, σ_yy and σ_zz act normal to their respective surfaces and hence are called normal stresses and the remaining independent components, σ_xy, σ_yz and σ_zx act parallel to the surface and hence are called shear stresses. (Remember that Cauchy stress is a symmetric tensor.) When the stress components are determined with respect to any coordinate basis vectors, there will be three normal stresses corresponding to σ_ii and three shear stresses, σ_ij , i ≠ j. Here to call certain components of the stresses as normal and others as shear, we have exploited the relationship between the components of the stress tensor and the traction vector.