Stress at a point—its implication in design
The state of stress at a point is given by nine stress components as shown in figure 126.96.36.199 and this is represented by the general matrix as shown below.
188.8.131.52F- Three dimensional stress field on an infinitesimal element.
Consider now a two dimensional stress element subjected only to shear stresses. For equilibrium of a 2-D element we take moment of all the forces about point A ( figure-184.108.40.206) and equate to zero as follows:
220.127.116.11F- Complimentary shear stresses on a 2-D element.
This gives τxy=τyx indicating that τxy and τyx are complimentary. On similar arguments we may write τyz=τzy and τzx=τxz . This means that the state of stress at a point can be given by six stress components only. It is important to understand the implication of this state of stress at a point in the design of machine elements where all or some of the stresses discussed above may act.
For an example, let us consider a cantilever beam of circular cross-section subjected to a vertical loading P at the free end and an axial loading F in addition to a torque T as shown in figure 18.104.22.168. Let the diameter of cross-section and the length of the beam be d and L respectively.
The maximum stresses developed in the beam are :
It is now necessary to consider the most vulnerable section and element. Since the axial and torsional shear stresses are constant through out the length, the most vulnerable section is the built-up end. We now consider the three elements A, B and C. There is no bending stress on the element B and the bending and axial stresses on the element C act in the opposite direction. Therefore, for the safe design of the beam we consider the stresses on the element A which is shown in figure 22.214.171.124.
Principal stresses and maximum shear stresses can now be obtained and using a suitable failure theory a suitable diameter of the bar may be obtained.