Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev

Machine Design

Mechanical Engineering : Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev

The document Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Machine Design.
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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.

Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev
 

Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev
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:

Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev
2.1.6.2F- Complimentary shear stresses on a 2-D element.


This gives τxyyx indicating that τxy and τyx are complimentary. On similar arguments we may write τyzzy and τzxxz . 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.

Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev

The maximum stresses developed in the beam are :

Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev

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.

Simple Stresses (Part - 3) Mechanical Engineering Notes | EduRev

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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