Whenever a machine component such as a shaft or structure is under loaded condition a complex stress is generated in its material. The machine Designer has to take care of the weakest part or portion, the maximum possible stress condition, or other practical factors for the component to survive under load conditions. It is critical to understand how the stress distribution takes place with the applied load.
When the force or load is applied on the material then every small element of the material can experience different stresses at the same time. For simplification and understanding of the concept of Stress (and Principal Stress), we will analyze the infinitesimal particle of a cube shape. there are 2 types of stress generated namely – Normal Stress and Shear Stress under loaded conditions.
Normal Stress is perpendicular to the surface which is either tensile or compressive. Imagine you press a surface with your hand! Shear Stress along the surface. Imagine a bread sandwich and you move the bread slices in opposite directions. Sandwich will be Sheared!
Consider a 2-dimensional shape, which means a single plane, Normal stress will have one component either tensile or compressive and Shear Stress will have 2 components in opposite directions because they act in pairs in opposite directions. So, a total of three components are in one plane.
So, for a 3-dimensional case, if we consider a case of a cube with 3 planes one is each X, Y, Z (or minus X, minus Y, and minus Z). A positive sign for one direction and a minus for the opposite direction. This means there will be a total of 9 stress components in a cube (3 in each plane).
Similarly, 3D and 2D strains will also have 9 and 3 components respectively.
Components σxx, σyy, and σzz are the normal stress and τxy, τyz, and τzx are the shear stress.
Principle Stress
For any combination of applied stress, there will be continuous distribution of the stress field around any point analyzed. The normal and shear stress at a point will vary with direction in any coordinate system chosen. However, despite the complex state of stress in a component, any design engineer will be interested in designing a component that is safe under given load conditions. That means, if the maximum stress at any point is below the safe value then the machine part will not fail. So we need to find that maximum normal and shear stress.
There will always be planes where shear stress is zero. The Normal stress on this plane is the Principle Normal Stress. Planes on which this principle stresses act are known as principle planes.
Similarly, There will be another plane on which shear stress will be maximum, The principal shear stress acts on a set of planes that are at 45 degrees angle to the planes of principle normal stress.