Stress on an Inclined Plane

The Cauchy stress tensor , a, at any point of a body (assumed to behave as a continuum) is completely defined by nine component stresses-three orthogonal, normal stresses and six orthogonal, shear stresses. It is used for the stress analysis of materials undergoing small deformations, in which the differences in stress distribution, in most cases, can be neglected. 

Recall that the directional cosines of a vector are often defined as being the cosines of the angles that the vector makes with the x, y and z axes, respectively. These angles are labeled a (the x axis angle), p (the y axis angle) and y (the z axis angle), while defining 1 = cosa, m = cos) and n = cosy. 

The sum of the forces acting on tetrahedron ABCP should be zero. The force components, in terms of their x, y and z directions, may be expressed as  

The forces in the x, y, z directions are obtained by multiplying the stresses by the areas upon which they act, giving these forces in their respective directions as in the x direction  

One can briefly summarize the above Eqs. (1.16b), (1.16d) and (1.16f) for the stresses at any point on the inclined plane by writing the stress tensor in any Cartesian coordinate system as  

---

Fig. 1.15 The normal stress on an inclined plane is resolved into its components, acting along the directions of the indicated coordinate system...

---