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**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 2.1.6.1 and this is represented by the general matrix as shown below.

**2.1.6.1F- 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-2.1.6.2) and equate to zero as follows:

**2.1.6.2F- Complimentary shear stresses on a 2-D element.**

This gives τ_{xy}=τ_{yx} indicating that τ_{xy} and τ_{yx} are complimentary. On similar arguments we may write τ_{yz}=τ_{zy} and τ_{zx}=τ_{xz} . 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 2.1.6.3. 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 2.1.6.4.

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.

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