Design Against Fluctuating Load MCQ Quiz - Objective Question with Answer for Design Against Fluctuating Load - Download Free PDF

Explanation:

Fatigue Testing and the S-N Curve

Definition: Fatigue testing is a method used to determine the durability and lifespan of a material under cyclic loading conditions. It is crucial in predicting how a material will behave when subjected to repeated stress or strain over time. The S-N curve, also known as the Wöhler curve, is a graphical representation that illustrates the relationship between the stress amplitude (S) and the number of cycles to failure (N).

Working Principle: During fatigue testing, a sample material is subjected to repeated cyclic loading until failure occurs. The stress amplitude and the number of cycles to failure are recorded. This data is then used to plot the S-N curve, where the x-axis represents the number of cycles to failure (N) on a logarithmic scale, and the y-axis represents the stress amplitude (S).

Correct Option Analysis:

The correct option is:

Option 2: The number of cycles and stress amplitude

This option accurately describes the S-N curve in fatigue testing. The S-N curve shows the relationship between the number of cycles to failure (N) and the stress amplitude (S). The curve typically displays how the material's endurance limit or fatigue limit is determined, which is the stress level below which the material can endure an infinite number of cycles without failing.

The S-N curve is critical for engineers and designers to understand the fatigue behavior of materials and to ensure the reliability and safety of components subjected to cyclic loading in various applications, such as in automotive, aerospace, and structural engineering

Explanation:

Notch sensitivity:

Notch sensitivity is defined as the susceptibility of a material to succumb to the damaging effect of stress raising notches in fatigue loading

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

Key Parameters:

• A = Notch Sensitivity Factor
• B = Theoretical Stress Concentration Factor (K_t)
• C = Fatigue Stress Concentration Factor (K_f)

actual stress = Kf σo, theoretical stress = Kt σo

where, σo = nominal stress, Kf = actual stress concentration factor K_t = theoretical stress concentration factor.

Increase of actual stress = (Kf σo - σo)

Increase of theoretical stress = (Kt σo - σo)

\({\bf{q}} = \frac{{\left( {{{\bf{K}}_{\bf{f}}}{{\bf{\sigma }}_{\bf{o}}}\; - \;{{\bf{\sigma }}_{\bf{o}}}} \right)}}{{\left( {{{\bf{K}}_{\bf{t}}}{{\bf{\sigma }}_{\bf{o}}}\; - \;{{\bf{\sigma }}_{\bf{o}}}} \right)}} = \frac{{\left( {{{\bf{K}}_{\bf{f}}}\; - \;1} \right)}}{{\left( {{{\bf{K}}_{\bf{t}}}\; - \;1} \right)}}\)

⇒ C = 1 + A [B - 1]

Explanation:

Fatigue Analysis: Goodman Line vs. Soderberg Line

• Fatigue analysis is essential in mechanical engineering to predict the life of a component under cyclic loading. Two common methods used in fatigue analysis are the Goodman line and the Soderberg line. These methods help engineers determine the safety and durability of materials subjected to fluctuating stresses.

Option 3: The Goodman line is based on the ultimate tensile strength, while the Soderberg line uses yield strength.

This option accurately describes the fundamental difference between the two lines used in fatigue analysis. The Goodman line incorporates the ultimate tensile strength (UTS) of the material, which is the maximum stress that a material can withstand while being stretched or pulled before breaking.
Soderberg line for ductile materials gives upper limit for any combination of mean and alternating stress. Following diagram depicts the same.

Considering following notations

σa = limiting safe stress amplitude

Se = endurance limit of the component

σm = limiting safe mean stress

Sut = ultimate tensile strength

Syt = Yield strength

N = Factor of safety

Soderberg line:

The line joining Syt (yield strength of the material) on the mean stress axis and Se (endurance limit of the component) on stress amplitude axis is called as Soderberg line. This line is used when yielding defines failure (Ductile materials).

The equation for the Soderberg line:

\(\frac{{{{\rm{\sigma }}_{\rm{m}}}}}{{{{\rm{S}}_{{\rm{yt}}}}}} + \frac{{{{\rm{\sigma }}_{\rm{a}}}}}{{{{\rm{S}}_{\rm{e}}}}} = \frac{1}{N}\)

Goodman line:  

Line joining Se on stress amplitude axis and Sut on mean stress axis is known as Goodman line. The triangular region below this line is considered a safe region. 
The equation for Goodman line:

\(\frac{{{\sigma _m}}}{{{S_{ut}}}} + \frac{{{\sigma _a}}}{{{S_e}}} = \frac{1}{N}\)

Explanation:

Soderberg line:
Soderberg line for ductile materials gives upper limit for any combination of mean and alternating stress. Soderberg line is the most conservative fatigue failure criterion. Following diagram depicts the same.

Considering following notations

σa = limiting safe stress amplitude

Se = endurance limit of the component

σm = limiting safe mean stress

Sut = ultimate tensile strength

Syt = Yield strength

N = Factor of safety

Soderberg line:

The line joining Syt (yield strength of the material) on the mean stress axis and Se (endurance limit of the component) on stress amplitude axis is called as Soderberg line. This line is used when yielding defines failure (Ductile materials).

The equation for the Soderberg line:

\(\frac{{{{\rm{\sigma }}_{\rm{m}}}}}{{{{\rm{S}}_{{\rm{yt}}}}}} + \frac{{{{\rm{\sigma }}_{\rm{a}}}}}{{{{\rm{S}}_{\rm{e}}}}} = \frac{1}{N}\)

Additional Information

Goodman line:

Line joining Se on stress amplitude axis and Sut on mean stress axis is known as Goodman line. The triangular region below this line is considered a safe region. 
The equation for Goodman line:

\(\frac{{{\sigma _m}}}{{{S_{ut}}}} + \frac{{{\sigma _a}}}{{{S_e}}} = \frac{1}{N}\)

Gerber Line: 

Line joining Se on stress amplitude axis and Sut on mean stress axis is joined by a parabolic curve. 
The equation for Gerber line:

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

Soderberg line is the most conservative fatigue failure criterion. Even yielding is not considered in soderberg criterion. 

The Gerber parabola is the best fit for the failure points of the test data.

The Goodman is safer from design consideration since it lies completely inside the Geber parabola and failure points.

Concept:

When a component is subjected to fluctuating stresses, there is mean stress (σ_mean) as well as stress amplitude (σ_amp).

To design those components certain methods are used based on the Endurance limit of the material σ_e.

When a straight-line joins Endurance limit σ_e on ordinate and Yield strength σ_yt on abscissa it is known as Soderberg line.

The abscissa represents σ_mean and ordinate represents σ_amp.

Soderberg Line is given by -

\(\frac{{{{\rm{\sigma }}_{{\rm{mean}}}}}}{{{{\rm{\sigma }}_{{\rm{yt}}}}}} + \frac{{{{\rm{\sigma }}_{{\rm{amp}}}}}}{{{{\rm{\sigma }}_{\rm{e}}}}} = \frac{1}{{{\rm{FOS}}}}\)

Calculation:

Given:

σ_max = 50 MPa, σ_min = 10 MPa, σ_yt = 300 MPa, σ_e = 100 MPa

\(\therefore {{\rm{\sigma }}_{{\rm{mean}}}} = \frac{{{{\rm{\sigma }}_{{\rm{max}}}} + {{\rm{\sigma }}_{{\rm{min}}}}}}{2} \Rightarrow \frac{{50 + 10}}{2} = 30{\rm{\;MPa}}\)

\(\therefore {{\rm{\sigma }}_{{\rm{amp}}}} = \frac{{{{\rm{\sigma }}_{{\rm{max}}}} - {{\rm{\sigma }}_{{\rm{min}}}}}}{2} \Rightarrow \frac{{50 - 10}}{2} = 20{\rm{\;MPa}}\)

Equation of Soderberg Line is given by –

\(\frac{{{{\rm{\sigma }}_{{\rm{mean}}}}}}{{{{\rm{\sigma }}_{{\rm{yt}}}}}} + \frac{{{{\rm{\sigma }}_{{\rm{amp}}}}}}{{{{\rm{\sigma }}_{\rm{e}}}}} = \frac{1}{{{\rm{FOS}}}}\)

\( \Rightarrow \frac{{30}}{{300}} + \frac{{20}}{{100}} = \frac{1}{{FOS}}\)

\( \Rightarrow 0.1 + 0.2 = \frac{1}{{FOS}}\)

\( \Rightarrow FOS = \frac{1}{{0.3}}\)

⇒ FOS = 3.33

Additional Information 

Goodman Line:

\(\frac{{{{\rm{\sigma }}_{{\rm{mean}}}}}}{{{{\rm{\sigma }}_{{\rm{ut}}}}}} + \frac{{{{\rm{\sigma }}_{{\rm{amp}}}}}}{{{{\rm{\sigma }}_{\rm{e}}}}} = \frac{1}{{{\rm{FOS}}}}\)

Gerber Line:

\({\left( {\frac{{{\rm{FOS}} \times {{\rm{\sigma }}_{{\rm{mean}}}}}}{{{{\rm{\sigma }}_{{\rm{ut}}}}}}} \right)^2} + \left( {\frac{{{\rm{FOS}} \times {{\rm{\sigma }}_{{\rm{amp}}}}}}{{{{\rm{\sigma }}_{\rm{e}}}}}} \right) = 1\)

Concept:

Stress Amplitude: It is the amount of stress which deviates from mean stress.

Stress Amplitude = (σ­­_max – σ_min)/2 

Stress Ratio: It is the ratio of the minimum stress experienced during a cycle to the maximum stress experienced during a cycle.

Stress Ratio = σ_min / σ_max

σ_max = Maximum stress experienced in a cycle

σ_min = Minimum stress experienced in a cycle

Calculation:

σ_max = 250 MPa σ_min = 50 MPa

\(\begin{array}{l} {\rm{Stress\ Amplitude}} = \frac{{{{\rm{σ }}_{{\rm{max}}}} - {{\rm{σ }}_{{\rm{min}}}}}}{2} = \frac{{250 - 50}}{2} = 100\ {\rm{MPa}}\\ {\rm{Stress\ Ratio}} = \frac{{{{\rm{σ }}_{{\rm{min}}}}}}{{{{\rm{σ }}_{{\rm{max}}}}}} = \frac{{50}}{{250}} = 0.2 \end{array}\)

Explanation:

The common form of presentation of fatigue data is by using the S-N curve, where the total cyclic stress (S) is plotted against the number of cycles to failure (N) in a logarithmic scale.

the fatigue life reduces with respect to the increase in stress range and at a limiting value of stress, the curve flattens off. The point at which the S-N curve flattens off is called the ‘endurance limit’.

The line between 103 and 106 cycles is taken to represent high cycle fatigue.

From the S-N curve, we can see that the curve becomes asymptotic nearly at 106 cycles.

Explanation:

Notch sensitivity:

Notch sensitivity is defined as the susceptibility of a material to succumb to the damaging effect of stress raising notches in fatigue loading

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

actual stress = K_f σ_o, theoretical stress = K_t σ_o

where, σo = nominal stress, K_f = actual stress concentration factor Kt = theoretical stress concentration factor.

Increase of actual stress = (K_f σ_o - σ_o)

Increase of theoretical stress = (K_t σ_o - σ_o)

\({\bf{q}} = \frac{{\left( {{{\bf{K}}_{\bf{f}}}{{\bf{\sigma }}_{\bf{o}}}\; - \;{{\bf{\sigma }}_{\bf{o}}}} \right)}}{{\left( {{{\bf{K}}_{\bf{t}}}{{\bf{\sigma }}_{\bf{o}}}\; - \;{{\bf{\sigma }}_{\bf{o}}}} \right)}} = \frac{{\left( {{{\bf{K}}_{\bf{f}}}\; - \;1} \right)}}{{\left( {{{\bf{K}}_{\bf{t}}}\; - \;1} \right)}}\)

Additional Information

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

The common form of presentation of fatigue data is by using the S-N curve, where the total cyclic stress (S) is plotted against the number of cycles to failure (N) in a logarithmic scale.

the fatigue life reduces with respect to the increase in stress range and at a limiting value of stress, the curve flattens off. The point at which the S-N curve flattens off is called the ‘endurance limit’.

The line between 103 and 106 cycles is taken to represent high cycle fatigue.

Explanation:

The endurance limit of the specimen is given by

S_e = K_a K_b K_c K_d S_e'

where K_a =  Surface finish factor, K_b = Size factor, K_c = Reliability factor, K_d = Modifying factor to account for stress concentration, S_e = Endurance limit stress of a particular mechanical component subjected to reversed bending stress (N/mm²), S_e' = Endurance limit the stress of a rotating beam specimen subjected to reversed bending stress (N/mm²)

• When the surface finish is poor, there are scratches and geometric irregularities on the surface. These surface scratches serve as stress raisers and result in stress concentration. The endurance limit is reduced due to the introduction of stress concentration on these scratches.
• When the machine part is larger greater is the probability that a flaw exists somewhere in the component, the chances of fatigue failure originating at these flaws are more. The endurance limit, therefore, reduces with the increasing size of the component.
• The greater the likelihood that a part will survive, the more is the reliability factor. The reliability factor K_c depends upon the reliability that is used in the design of the component.

∴ The endurance limit is reduced due to stress concentration.

Explanation:

Fluctuating load:

When the magnitude of forces varies with time. Due to this varying load, varying stresses induced in a material. They are of three types-

• Fluctuating stress
• Repeated stress
• Reversed stress

It is observed that 80 % of the failure of a mechanical component is due to fatigue failure resulting from fluctuating stresses.

Fatigue failure:

• It is defined as a time-delayed fracture under cycling loading.
• They are sudden and total.
• It depends upon the number of cycles, mean stress, stress amplitude, stress concentration residual stress.

Endurance Limit:

• The maximum amplitude of a completely reversed stress that the standard specimen can sustain for unlimited cycles without fatigue failure.
• Since the fatigue test cannot be conducted for an unlimited or infinite number of cycles, 10⁶ cycles are considered as infinite life of a material.

Additional Information

Components subjected to Low-cycle-fatigue (up to 10³ cycles) are designed on the basis of Ultimate/Yield strength with a factor of safety (FOS).

Components subjected to High-cycle-fatigue (103 - 10⁶ cycles) are designed on the basis of the Endurance limit with a factor of safety (FOS).

Explanation:

Endurance limit (σe):

It is defined as the maximum value of completely reversed bending stress which a polished standard specimen can withstand without failure, for an infinite number of cycles (usually 106 cycles).

Hence for steel in fatigue loading the endurance limit is the maximum reversed bending stress it can withstand without failure for an infinite number of cycle.

Additional Information

There are various factors on which endurance limits depends which changes the value of endurance limit by some factors.

i.e. σe’ = σe × (Ka × Kb × Kc × Km)

Surface finish factor (Ka): The endurance limit of the specimen depends on the surface conditions.

• If the surface is smooth, the value of Ka is 1.
• If crack increases and surface become rough, the value of Ka decreases.

Size factor (Kb): If the size of the standard specimen is increased. Then the endurance limit of the material will decrease. This is because a longer specimen will have more defects than a smaller one.

Load factor (Kc): The endurance limit changes as the type of loading changes.

• For a completely reverse bending load → Kc = 1
• For a completely reverse axial load → Kc = 0.85
• For a completely reverse shear load → Kc = 0.5

Miscellaneous factor (Km): Apart from above factors there are other miscellaneous factors on which endurance limit depends such as reliability factor (Kr), temperature factor (K­­t), impact factor (Ki) etc.