Option 4 : 1

__Explanation:__

Notch Sensitivity (q):

\({\rm{q}} = \frac{{{\rm{Increase\;of\;fatigue\;stress\;over\;nominal\;stress}}}}{{{\rm{Increase\;of\;theoretical\;stress\;over\;nominal\;stress}}}} = \frac{{{{\rm{k}}_{\rm{f}}} - 1}}{{{{\rm{k}}_{\rm{t}}} - 1}}\)

Where,

Theoretical Stress Concentration Factor (Kt):

It is the ratio of actual stress near discontinuity to the nominal stress from the elementary equation.

\({{\rm{k}}_{\rm{t}}} = \frac{{{\rm{Highest\;value\;of\;the\;actual\;stress\;near\;discontinuity}}}}{{{\rm{Nominal\;stress\;calculated\;from\;elementary\;equation}}}} = \frac{{{\sigma _{max}}}}{{{\sigma _o}}} = \frac{{{\tau _{max}}}}{{{\tau _o}}}\)

Fatigue Stress Concentration Factor (Kf):

It is the ratio of the fatigue strength of an unnotched member to the fatigue strength of the same member with a notch.

\({{\rm{k}}_{\rm{f}}} = \frac{{{\rm{Effective\;fatigue\;stress}}}}{{{\rm{Nominal\;fatigue\;stress}}}}\)

q = 0 ⇒ k_{f} = 1 → **Notch is not sensitive in fatigue.**

q = 1 ⇒ k_{f} = k_{t} → **Notch is fully sensitive in fatigue.**

Option 3 : 1.1

__Concept:__

**Theoretical Stress Concentration Factor (K _{t}):**

It is the ratio of actual stress near discontinuity to the nominal stress from the elementary equation.

\({{\rm{K}}_{\rm{t}}} = \frac{{{\rm{Highest\;value\;of\;the\;actual\;stress\;near\;discontinuity}}}}{{{\rm{Nominal\;stress\;calculated\;from\;elementary\;equation}}}} = \frac{{{\sigma _{max}}}}{{{\sigma _o}}} = \frac{{{\tau _{max}}}}{{{\tau _o}}}\)

**Fatigue Stress Concentration Factor (K _{f}):**

It is the ratio of the fatigue strength of an unnotched member to the fatigue strength of the same member with a notch.

\({{\rm{K}}_{\rm{f}}} = \frac{{{\rm{Effective\;fatigue\;stress}}}}{{{\rm{Nominal\;fatigue\;stress}}}}\)

**Notch Sensitivity (q):**

\({\rm{q}} = \frac{{{\rm{Increase\;of\;fatigue\;stress\;over\;nominal\;stress}}}}{{{\rm{Increase\;of\;theoretical\;stress\;over\;nominal\;stress}}}} = \frac{{{{\rm{K}}_{\rm{f}}} - 1}}{{{{\rm{K}}_{\rm{t}}} - 1}}\)

__Calculation:__

__Given:__

K_{t} = 2, q = 0.1

∵ \(q = \frac{{{{\rm{K}}_{\rm{f}}} - 1}}{{{{\rm{K}}_{\rm{t}}} - 1}}\)

\( \Rightarrow 0.1 = \frac{{{K_f} - 1}}{{2 - 1}}\)

⇒ K_{f} – 1 = 0.1

⇒ **K _{f} = 1.1**

Theoretical stress factor (K_{t}) can also be calculated from the figure if irregularities dimension is given-

\({K_t} = 1 + 2\left( {\frac{A}{B}} \right)\)

where a = Semi-major axis perpendicular to the direction of load and b = semi-minor axis parallel to the direction of load.

**For sharp crack**

B → 0 ∴ K_{t} = ∞

**For circle**

A = B

∴ K_{t} = 3

Option 3 : increase in the surface roughness and increase in the size of the beam

**Explanation:**

Corrected endurance strength is defined as

σ_{e} = K_{a} K_{b} K_{c} K_{d}.σ^{’}_{e}

where, σ^{’}_{e} = Endurance strength and K_{a} = size factor, K_{b} = surface factor, K_{c} = load factor, K_{d} = Temperature factor

{K_{a}, K_{b}, K_{c}, K_{d}} < 1

Option 2 : 50 MPa

**Concept:**

**Stress developed at irregular section = Normal average stress × Stress concentration factor**

**Stress developed at irregular section ≤ maximum permissible stress**

**Calculation:**

**Given:**

Stress concentration factor = 3 , maximum permissible stress = 150 MPa

Stress developed at irregular section ≤ maximum permissible stress

∴ Normal average stress × stress concentration factor ≤ 150

Normal average stress × 3 = 150

**∴ Normal average stress = 50 MPa **

For a plate with an elliptical hole subjected to tensile stress, σ as shown in the figure, the maximum tensile stress at point P is

Option 4 : More than 3σ

**Concept:**

Maximum Tensile stress at point P is \(= {\sigma _0}\left( {1 + \frac{{2a}}{b}} \right)\)

where \(\left( {1 + \frac{{2a}}{b}} \right)\) is Theoretical stress concentration factor.

From figure it is clear that a > b

\(\therefore 1 + \frac{{2a}}{b} > 3\)

Hence maximum tensile stress in more than 3σ.

If a = b then the stress at point P will be equal to 3 times the applied stress.Option 3 : stress concentration

**Explanation:**

Stress concentration:

- Stress concentration is defined as the
**localization of high stresses**due to irregularities present in the component and**abrupt changes in the cross-section.**

**Causes of stress concentration:**

**Variation in properties of the material: **

In general, the material is no homogenous throughout, there are some variations in the material properties due to the following factors:

- Internal cracks and flaws like blowholes.
- Cavities in welds.
- Air holes in steel components.
- Non-metallic inclusions.

These variations act as discontinuities in the component and cause stress concentration.

Load application: The machine components are **subjected to forces**. These forces act either at a small or point area on the component. Since the area is small, the pressure at these points is excessive. This results in stress concentration.

Abrupt changes in section: The abrupt changes are due to steps cut on the shafts to accommodate the **bearings, pulleys sprockets**. These create change in the cross-section of the shaft ad results in the stress concentration.

Discontinuities in the component: There are some features of machine components such as oil holes, keyways, and splines, and screw threads result in a discontinuity in the cross-section of the component. There is stress concentration in the vicinity of these discontinuities.

Machining Scratches: Machining scratches stamp marks or inspection marks are surface irregularities that cause stress concentration.

Option 4 : Providing fastener

**Explanation:**

**Reduction of stress concentration**:

Although it is not possible to completely eliminate the effect of stress concentration, there are methods to reduce stress concentration. They are:

**Additional notches and holes in tension member**:

A single notch results in a high degree of stress concentration. The impact is reduced by three methods:

- use of multiple notches
- drilling additional holes
- removal of undesired material

**Fillet radius, undercutting, and notch for members in bending**:

**Drilled additional holes in shaft**:

**Addition of washer**:

Simple washers are thin annular-shaped metallic disks. Its functions are as follows:

- Distributes the load over a large area on the surface of clamped parts.
- It provides bearing surfaces over large clearance holes.

Option 4 : 1

__Explanation:__

Notch Sensitivity (q):

\({\rm{q}} = \frac{{{\rm{Increase\;of\;fatigue\;stress\;over\;nominal\;stress}}}}{{{\rm{Increase\;of\;theoretical\;stress\;over\;nominal\;stress}}}} = \frac{{{{\rm{k}}_{\rm{f}}} - 1}}{{{{\rm{k}}_{\rm{t}}} - 1}}\)

Where,

Theoretical Stress Concentration Factor (Kt):

It is the ratio of actual stress near discontinuity to the nominal stress from the elementary equation.

\({{\rm{k}}_{\rm{t}}} = \frac{{{\rm{Highest\;value\;of\;the\;actual\;stress\;near\;discontinuity}}}}{{{\rm{Nominal\;stress\;calculated\;from\;elementary\;equation}}}} = \frac{{{\sigma _{max}}}}{{{\sigma _o}}} = \frac{{{\tau _{max}}}}{{{\tau _o}}}\)

Fatigue Stress Concentration Factor (Kf):

It is the ratio of the fatigue strength of an unnotched member to the fatigue strength of the same member with a notch.

\({{\rm{k}}_{\rm{f}}} = \frac{{{\rm{Effective\;fatigue\;stress}}}}{{{\rm{Nominal\;fatigue\;stress}}}}\)

q = 0 ⇒ k_{f} = 1 → **Notch is not sensitive in fatigue.**

q = 1 ⇒ k_{f} = k_{t} → **Notch is fully sensitive in fatigue.**

Option 2 : 204

__Concept:__

The notch sensitivity factor is given by:

\(q = \frac{{{K_f} - 1}}{{{K_t} - 1}} \Rightarrow {K_f} = 1 + q\left( {{K_t} - 1} \right)\;where,\;{K_f} = fatigue\;stress\;conc.\;factor\)

The modifying factor to account for stress concentration \({K_d} = \frac{1}{{{K_f}}}\)

The corrected endurance limit is: \({S_e}' = {S_e}{K_d} = \frac{{{S_e}}}{{{K_f}}}\)

__Calculation:__

\({K_f} = 1 + q\left( {{K_t} - 1} \right) = 1 + 0.8\left( {2.51 - 1} \right) = 2.208\)

\({S_e}' = \frac{{{S_e}}}{{{K_f}}} = \frac{{450}}{{2.208}} = 203.8\;MPa\)Option 3 : 1.1

__Concept:__

**Theoretical Stress Concentration Factor (K _{t}):**

\({{\rm{K}}_{\rm{t}}} = \frac{{{\rm{Highest\;value\;of\;the\;actual\;stress\;near\;discontinuity}}}}{{{\rm{Nominal\;stress\;calculated\;from\;elementary\;equation}}}} = \frac{{{\sigma _{max}}}}{{{\sigma _o}}} = \frac{{{\tau _{max}}}}{{{\tau _o}}}\)

**Fatigue Stress Concentration Factor (K _{f}):**

\({{\rm{K}}_{\rm{f}}} = \frac{{{\rm{Effective\;fatigue\;stress}}}}{{{\rm{Nominal\;fatigue\;stress}}}}\)

**Notch Sensitivity (q):**

\({\rm{q}} = \frac{{{\rm{Increase\;of\;fatigue\;stress\;over\;nominal\;stress}}}}{{{\rm{Increase\;of\;theoretical\;stress\;over\;nominal\;stress}}}} = \frac{{{{\rm{K}}_{\rm{f}}} - 1}}{{{{\rm{K}}_{\rm{t}}} - 1}}\)

__Calculation:__

__Given:__

K_{t} = 2, q = 0.1

∵ \(q = \frac{{{{\rm{K}}_{\rm{f}}} - 1}}{{{{\rm{K}}_{\rm{t}}} - 1}}\)

\( \Rightarrow 0.1 = \frac{{{K_f} - 1}}{{2 - 1}}\)

⇒ K_{f} – 1 = 0.1

⇒ **K _{f} = 1.1**

Theoretical stress factor (K_{t}) can also be calculated from the figure if irregularities dimension is given-

\({K_t} = 1 + 2\left( {\frac{A}{B}} \right)\)

where a = Semi-major axis perpendicular to the direction of load and b = semi-minor axis parallel to the direction of load.

**For sharp crack**

B → 0 ∴ K_{t} = ∞

**For circle**

A = B

∴ K_{t} = 3