Stress Concentration MCQ Quiz in मल्याळम - Objective Question with Answer for Stress Concentration - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Explanation:

Stress concentration:

• The concept of stress concentration originates from the observation that the stress at or near the periphery of a stress raiser rises dramatically in comparison to the nominal or average stress in that section.

• A stress raiser in a member can be: holes in a plate or bar, shouldered or stepped shafts, grooved shafts, keyways and splines, Bolt heads, screw threads and so on.

1. Theoretical Stress Concentration Factor (Kt):

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

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

2. 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.

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

Few points to keep in mind while dealing with the stress concentration factor are:

• For ductile materials under static loading, stress concentration can be ignored. In ductile materials under static load, small plastic deformation occurs near the yielding point and therefore redistribution of stresses take place. Because the plastic deformation is limited to a smaller area, no visible damage occurs.
• For brittle materials under static loading, stress concentration must be considered.
• For all materials subjected to dynamic loading, stress concentration must be considered.

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:

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

Concept:

Stress concentration due to holes:

\({k_t} = 1 + 2\left( {\frac{a}{b}} \right)\)

\(\begin{array}{l} {\sigma _{max}} = {k_t} \times {\sigma _0},{\rm{ }}A = \left( {b-d} \right)t\\ {\sigma _o} = \frac{{{\sigma _{max}}}}{{{k_t}}} = \frac{{75}}{3} = 25\;MPa\\ {\sigma _o} = \frac{P}{A} \Rightarrow \frac{{12 \times {{10}^3}}}{{\left( {60 - 12} \right)t}} = 25\\ t = \frac{{12 \times {{10}^3}}}{{48 \times 25}} = 10\;mm \end{array}\)

Note: Any such discontinuity in a member affects the stress distribution in the neighbourhood and the discontinuity acts as a stress raiser. There will be a sharp rise in stress in the vicinity of the hole.

In this question, there is a hole in the plate so we have to consider the stress concentration factor because in the question maximum stress is given.

Concept: 

Notch sensitivity is defined as the susceptibility of a material to succumb to the damaging effects of stress gassing notches in fatigue loading. The notch sensitivity factor a is defined as

\(q=\frac{increase~of~actual~stress~over~nominal~stress}{increase~of~theoretical~stress~over~nominal~stress}\)

The theoretical Stress concentration factor

\({{K}_{F}}={{\left( \frac{Endurance~limit~of~the~notch~free~specimen}{Endurance~limit~of~the~motched~specimen} \right)}_{actual}}\)

\({{K}_{t}}={{\left( \frac{Endurance~limit~of~the~notch~free~specimen}{Endurance~limit~of~the~notched~specimen} \right)}_{Theoretical}}\)

σ0 = Nominal stress as obtained by elementary equations

Actual stress = Kfσ0

Theoretical stress = Ktσ0

Increase of actual stress over nominal stress = (Kfσ0 - σ0)

Increase of Theoretical stress over nominal stress = (Ktσ0 - σ0)

\(q=\frac{\left( {{k}_{f}}{{\sigma }_{0}}-{{\sigma }_{0}}~ \right)}{\left( {{k}_{t}}{{\sigma }_{0}}-{{\sigma }_{0}}~ \right)}=\frac{{{k}_{f}}-1}{{{k}_{t}}-1~}\)

Calculation:

Given: k_t = 1

\(q=\frac{{{k}_{f}}-1}{{1}-1~}\)

⇒ kf = 1

Notch sensitivity is defined as the susceptibility of a material to succumb to the damaging effects of stress raising notches in fatigue loading. The notch sensitivity factor is defined as

\(q=\frac{increase~of~actual~stress~over~nominal~stress}{increase~of~theoretical~stress~over~nominal~stress}\)

The theoretical stress concentration factor

\({{K}_{t}}={{\left( \frac{Endurance~limit~of~the~notch~free~specimen}{Endurance~limit~of~the~notched~specimen} \right)}_{Theoretical}}\)

• σ0 = Nominal stress as obtained by elementary equations
• Actual stress = Kfσ0
• Theoretical stress = Ktσ0
• Increase of actual stress over nominal stress = (Kfσ0 - σ0)
• Increase of Theoretical stress over nominal stress = (Ktσ0 - σ0)

\(q=\frac{\left( {{K}_{f}}{{\sigma }_{0}}-{{\sigma }_{0}}~ \right)}{\left( {{K}_{t}}{{\sigma }_{0}}-{{\sigma }_{0}}~ \right)}=\frac{{{K}_{f}}-1}{{{K}_{t}}-1~}\)

The notch sensitivity factor is given by:

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

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. 

Concept:

From the theory of elasticity, the theoretical stress concentration factor is given by

\({K_t} = 1 + 2\left( {\frac{a}{b}} \right)\)

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

\(\Rightarrow {K_t} = 1 + 2 \times 2\)

\(\Rightarrow {K_t} = 5\)

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.

Concept:

The stress concentration factor (K) is defined as the ratio of the maximum stress in a member at discontinuity to the nominal or average stress at the same section based upon net area.

K or Kt = σmax/σ0 = τmax/τ0   where σ0 and τ0 are nominal stress and σmax and τmax are localized stresses at the discontinuity.  

Mathematically \(\rm K_t = 1\;+\;2(\frac{A}{B})\)

Stress concentration factor K is used for consideration of stress concentration and to find out the localized stress at the discontinuity.

The value of Kt depends upon the material and geometry of the part.

Calculation:

Given:​ \(\frac{A}{B} = 2\)

The Stress concentration factor (Kt) is dependent in loading,

If loading is along x the direction,

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

if loading along the y direction,

\({K_t} = 1 + \frac{{2A}}{B} = 1 + 2 × 2 = 5\)

Stress concentration factor (K) is defined as the ratio of the maximum stress in a member at discontinuity to the nominal or average stress at the same section based upon net area.

K or Kt = σmax/σ0 = τmax/τ0   where, σ0 and τ0 are nominal stress and σmax and τmax are localized stresses at discontinuity.  

Mathematically Kt = 1 + 2.a/b  

Stress concentration factor K is used for consideration of stress concentration and to find out the localized stress at the discontinuity.

The value of Kt depends upon the material and geometry of the part.