Learning Objectives:

• Able to formulate and analyze stresses and strains in machine elements and structures subjected to axial and shear loads.

• Ability to define and apply Factor of Safety (FOS) while designing a machine component.

1. Axial Stresses:
   
   

2. Axial forces are those whose line of action matches with the axis of the body. Depending upon the direction of the force, axial forces can be tensile force or compressive force. \[\sigma = \dfrac{F}{A}\] where A is the area of cross section perpendicular to the applied force direction.

3. Shear Stress: Shear stresses are created when the line of action of the forces on opposite side of a body is NOT in a single line. Shear forces can also be created in bending as well as in torsion.
   Shear stresses which tend to rotate an element clockwise, are considered positive. Normally a double subscripted notation is used to denote shear stresses depending upon which plane they are acting. For example, \(\tau_{xy}\) means the shear stress is acting on x-y plane.
   The formula for calculating the shear stress is similar to that for calculating axial stress, except with one major difference. The area which is under consideration in both formulae is totally different. While calculating stresses, the area which is resisting those forces is considered. In axial stresses, the area is area of cross-section perpendicular to the applied force direction, whereas in shear stresses it is parallel to the applied force direction. \[\tau = \dfrac{\mathrm{Shearing\ Force}}{\mathrm{Area\ in\ Shear}} = \dfrac{F}{A_s}\]
   
   

4. Torsional shear stress occurs when the outer surface of a solid round shaft is subjected to torque, it experiences the greatest shearing strain and the largest torsional shear stress. \[\tau_{max} = \dfrac{Tc}{J}\] Where
   c = radius of the shaft to its outside surface,
   J = polar moment of inertia

Solved Example: 89-1-01

Residual stress in materials:

A. Acts when external load is applied

B. Becomes zero when external load is removed

C. Is independent of external loads

D. Is always harmful

Residual stresses are stresses that remain in a solid material after the original cause of the stresses has been removed. Residual stress may be desirable or undesirable.

Correct Answer: C

Solved Example: 89-1-02

Stress concentration is caused due to:

A. Variation in properties of material from point to point in a member

B. Pitting at points or areas at which loads on a member are applied

C. Abrupt change of section

D. All of the above

Correct Answer: D

Solved Example: 89-1-03

The unit of modulus of elasticity is same as those of:

A. Stress, strain and pressure

B. Stress, force and modulus of rigidity

C. Strain, force and pressure

D. Stress, pressure and modulus of rigidity

Stress, pressure and modulus of rigidity all have units Pa, which is same as units of modulus of elasticity.

Correct Answer: D

Solved Example: 89-1-04

The stress induced in a body, when suddenly loaded, is ________ the stress induced when the same load is applied gradually. (KPSC JE 2016)

A. Equal to

B. One-half

C. Twice

D. Four times

Suddenly applied load causes twice the stress as compared to when the same load is applied gradually. For gradually applied load, $\sigma = \dfrac{F}{A}$ but for suddenly applied load, $\sigma = \dfrac{2F}{A}$

Correct Answer: C

Solved Example: 89-1-05

The ultimate strength of steel in tension in comparison to shear is in the ratio of:

A. 1:1

B. 2:1

C. 3:2

D. 2:3

Correct Answer: C

Solved Example: 89-1-06

Which is correct statement?

A. A member made of steel will generally be more rigid than a member of equal load-carrying ability made of cast iron.

B. A member made of cast iron will generally be. more rigid than a member of equal load carrying ability made of steel.

C. Both will be equally rigid.

D. Which one is rigid will depend on several other factors.

Correct Answer: B

When a structural member or a machine component is applied with some forces, not only resisting forces are created inside that component, but also it undergoes various levels of deformation. Excessive deformation are considered failure and they can severely affect the resisting abilities of a machine or structure.
We define strain as a quantity to study this in details.
Strain is defined as ratio of change in length to original length. \[\epsilon = \dfrac{\Delta L}{L}\]

Solved Example: 89-2-01

The deformation per unit length is called: (KPSC JE 2016)

A. Tensile stress

B. Compressive stress

C. Shear stress

D. Strain

Correct Answer: D

Solved Example: 89-2-02

A circular hollow tube made of steel is used to support a compressive load of 500kN. The inner and outer diameters of the tube are 90mm and 130mm respectively and its length is 1000mm. Due to compressive load, the contraction of the rod is 0.5mm. Determine the compressive stress and strain in the post.

A. $\sigma$ = -72.3 MPa, $\epsilon$ = -5 $\times$ $10^{-4}$

B. $\sigma$ = -56.1 MPa, $\epsilon$ = -5 $\times$ $10^{-4}$

C. $\sigma$ = -32.2 MPa, $\epsilon$ = -5 $\times$ $10^{-4}$

D. $\sigma$ = -88.4 MPa, $\epsilon$ = -5 $\times$ $10^{-4}$

\[\sigma = \dfrac{F}{A} = \dfrac{500\times 10^3}{\dfrac{\pi}{4}(130^2-90^2)}\times 10^{-6} = 72.3MPa\] \[\epsilon = - \dfrac{\Delta L}{L} = - \dfrac{0.5}{1000} = 5 \times 10^{-4}\]

Correct Answer: A