Soderberg Curve MCQ Quiz in मल्याळम - Objective Question with Answer for Soderberg Curve - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Concept:

Soderberg   Criterion	\(\frac{{{\sigma _m}}}{{{\sigma _{yt}}}} + \frac{{{\sigma _a}}}{{{\sigma _e}}} = \frac{1}{{FS}}\)
Goodman  Criterion	\(\frac{{{\sigma _m}}}{{{\sigma _{ut}}}} + \frac{{{\sigma _a}}}{{{\sigma _e}}} = \frac{1}{{FS}}\)
Gerber  Criterion	\({\left( {\frac{{{\sigma _m}}}{{{\sigma _{ut}}}}} \right)^2}FS + \frac{{{\sigma _a}}}{{{\sigma _e}}} = \frac{1}{{FS}}\)

Goodman Criterion:

\(\frac{{{\sigma _m}}}{{{\sigma _{ut}}}} + \frac{{{\sigma _a}}}{{{\sigma _e}}} \le 1\)

Calculation:

σ = 120 + k sin (20 t + 0.7)

σ_max = 120 + K (when sin (20t +0.7) = 1) 

σ_min = 120 – K (when sin (20t +0.7) = – 1) 

\(\therefore {\sigma _{mean}} = \frac{{{\sigma _{max}} + {\sigma _{min}}}}{2} = \frac{{120 + K + 120 - K}}{2} = 120\;MPa\)

\({\sigma _a} = \frac{{{\sigma _{max}} - {\sigma _{min}}}}{2} = \frac{{120 + K - \left( {120 - K} \right)}}{2} = K\)

So for safe design

\(\frac{{120}}{{360}} + \frac{K}{{210}} \le 1\)

⇒ K ≤ 140

Explanation:

When designing mechanical components that are subjected to dynamic loading, it is crucial to assess the risk of fatigue failure. Fatigue failure occurs due to the repeated application of loads, which can cause a material to fail at stress levels lower than its ultimate tensile strength. One of the methods used to assess fatigue failure is the Goodman line, which provides a criterion for safe design based on the mean and alternating stresses experienced by the component.

The Goodman line is an empirical relationship that helps engineers determine whether a component will fail under a given set of loading conditions. It is particularly useful for components subjected to cyclic loading, where both mean stress (\(\sigma_m\)) and alternating stress (\(\sigma_a\)) are present.

The Goodman line can be represented mathematically by an equation that relates the mean stress, alternating stress, and the material properties of the component. The general form of the Goodman equation is:

\(\frac{K_f \sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n_s}\)

Where:

• K_f = Fatigue stress concentration factor
• S_e = Modified endurance limit
• S_{ut} = Ultimate tensile strength
• \(\sigma_m\) = Mean stress
• \(\sigma_a\) = Alternating stress
• n_s = Factor of safety

This equation is used to evaluate the safety of a component under cyclic loading. If the left-hand side of the equation is less than or equal to the right-hand side, the component is considered safe from fatigue failure under the given loading conditions.

Solution:

Given the options in the question, we need to identify which expression correctly represents the Goodman line for assessing fatigue failure.

Option 1: \(\frac{K_f n_s \sigma_a}{S_e} + \frac{n_s \sigma_m}{S_{ut}} = 1\)

This option incorrectly includes the factor of safety \(n_s\) in both terms, which is not consistent with the standard Goodman equation.

Option 2: \(\frac{K_f \sigma_a}{S_e} + \frac{\sigma_m}{S_{yt}} = \frac{\sigma_m}{n_s}\)

This option incorrectly uses the yield strength \(S_{yt}\) instead of the ultimate tensile strength \(S_{ut}\) and also misplaces the factor of safety \(n_s\).

Option 3: \(\frac{K_f \sigma_a}{S_e} + \left(\frac{n_s \sigma_m}{S_{ut}}\right)^2 = \frac{1}{n_s}\)

This option incorrectly squares the term involving the mean stress and factor of safety, which is not consistent with the standard Goodman equation.

Option 4: \(\frac{K_f \sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n_s}\)

This option correctly represents the Goodman line. It includes the fatigue stress concentration factor \(K_f\), modified endurance limit \(S_e\), ultimate tensile strength \(S_{ut}\), mean stress \(\sigma_m\), alternating stress \(\sigma_a\), and factor of safety \(n_s\) in the correct form.

Therefore, the correct answer is option 4:

\(\frac{K_f \sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n_s}\)

This equation is used to assess the safety of a mechanical component under cyclic loading conditions. It ensures that the combined effect of mean and alternating stresses does not exceed the material's capacity to withstand fatigue, considering the factor of safety.

Concept:

Mean or average stress:

\({\sigma _m} = \frac{{{\sigma _{max}} + {\sigma _{min}}}}{2}\)

Reversed stress component or alternating or variable stress:

\(\sigma = \frac{{{\sigma _{max}} - {\sigma _{min}}}}{2}\)

Max bending Moment for simply supported Beam at centre: \(M = \frac{{PL}}{4}\)

Calculation:

\(\begin{array}{l} Z = \frac{\pi }{{32}}{d^3} = \frac{\pi }{{32}}{\left( {60} \right)^3} = 21205.75m{m^3}\\ {P_m} = \frac{{{P_{max}} + {P_{min}}}}{2} = \frac{{4P + P}}{2} = 2.5P\\ {P_a} = \frac{{{P_{max}} - {P_{min}}}}{2} = \frac{{4P - P}}{2} = 1.5P \end{array}\)

Bending Moment:

\(\begin{array}{l} {M_m} = \frac{{{P_m} \times L}}{4}\\ {M_a} = \frac{{{P_a} \times L}}{4} \end{array}\)

Bending Stress:

\(\begin{array}{l} {\sigma _m} = \frac{{{M_m}}}{Z} = \frac{{{P_m} \times L}}{{4Z}} = \frac{{2.5P \times 500}}{{4 \times 21205.75}} = 14.74 \times {10^{ - 3}}P\;MPa\\ {\sigma _a} = \frac{{{M_a}}}{Z} = \frac{{{P_a} \times L}}{{4Z}} = \frac{{1.5P \times 500}}{{4 \times 21205.75}} = 8.84 \times {10^{ - 3}}P\;MPa \end{array}\)

Soderberg Equation:

\(\begin{array}{l} \frac{1}{{F.S}} = \frac{{{\sigma _m}}}{{{\sigma _y}}} + \frac{{{\sigma _a}}}{{{\sigma _e}}}\\ \frac{1}{{1.4}} = P\left( {\frac{{14.74 \times {{10}^{ - 3}}}}{{390}} + \frac{{8.84 \times {{10}^{ - 3}}}}{{260}}} \right) = 71.79 \times {10^{ - 6}}P \end{array}\)

P = 9949 N = 9.9 kN

Explanation:

Von - mises yield criterion: This criterion is based on the determination of the distortion energy in a given material. It is used to find failure stress in the ductile materials.

Soderberg’s Law: It is used for the component subjected to fluctuating load for fatigue design.

Sommer field Number: It is a dimensionless quantity used in the design of hydrodynamic journal bearings. It is very important in lubrication analysis because it contains all design variables.

Buckingham equation: This equation gives the durability of gears and the dynamic loads to which they are subjected.

Explanation:

Endurance Limit:

• The endurance limit (also known as the fatigue limit) is the maximum stress amplitude below which a material can endure an essentially infinite number of load cycles without failing.
• For many materials, particularly ferrous alloys (like steel), there is a well-defined endurance limit.
• If the stress amplitude is kept below this limit, the material theoretically will not fail due to fatigue, regardless of the number of cycles.

Importance of Endurance Limit in Shaft Design:

When designing a shaft for fluctuating loads, the endurance limit is crucial because:

• Preventing Fatigue Failure: By ensuring that the fluctuating stresses in the shaft are below the endurance limit, designers can prevent fatigue failure, which is vital for the safe and reliable operation of machinery.
• Long-Term Performance: Components designed with the endurance limit in mind can withstand a large number of cycles, ensuring long-term performance and durability.
• Safety: Fatigue failure can be sudden and catastrophic. Designing below the endurance limit enhances the safety of mechanical systems.

In summary, the endurance limit is the primary consideration when designing a shaft for fluctuating loads because it directly relates to the material's ability to resist fatigue failure over an infinite number of cycles.

The Goodman line equation is used in mechanical engineering to predict the failure of materials under varying loads. It combines the effects of mean stress and alternating stress to determine the endurance limit of the material. The equation accounts for the ultimate tensile strength (Su') and the estimated actual endurance strength (S'n') of the material. By plotting the mean stress and alternating stress on a graph, the Goodman line helps in assessing whether the material will withstand the applied cyclic loads without failure.

Concept:

Soderberg Line - 

• When a component is subjected to fluctuating stresses, there is mean stress (S_m) and amplitude stress (S_a) play an important role. To design these components certain principles/methods are used based on the Endurance Limit of the material (S_e).
• The main methods for designing objects for fluctuating loads are:
  1. Soderberg Line Method
  2. Goodman Line Method
  3. Gerber Line Method
• When a straight line joins Endurance Limit on the y-axis and Yield Strength (S_yt) on the x-axis, then it becomes Soderberg Line.
• The y-axis represents amplitude stress and the x-axis represents mean stress.
• Soderberg Line is the most Conservative Method for designing objects on the basis of fluctuating stresses. 
• Soderberg Line is suitable for ductile materials.​

The equation of the Soderberg Line is given by:

\(\frac{S_a}{S_e} + \frac{S_m}{S_{yt}} = \frac{1}{FOS}\)

Mean Stress, S_m = \(\frac{\sigma_{max} + \sigma_{min}}{2}\)   

Amplitude Stress, S_a = \(\frac{\sigma_{max} - \sigma_{min}}{2}\)

Calculation:

Given:

\(\sigma _{max} = +300 \) MN/m², \(\sigma _{min} = -150 \) MN/m2, S_yt = 0.55S_u, Se = 0.5Su, FOS = 2

\(S_m = \frac{300-150}{2} = 75\) MN/m2

\(S_a = \frac{300+150}{2} = 225\) MN/m2

\(\frac{225}{0.5S_u} + \frac{75}{0.55S_u} = \frac{1}{2}\)

\(\frac{450}{S_u} + \frac{136.36}{S_u} = \frac{1}{2}\)

\(586.36 \times 2 = S_u\)

S_u = 1172.72  MN/m2  \(\simeq\) 1173  MN/m2

Concept:

For reversed loading |ω_max| = |ω_min|

\({\sigma _m} = mean\;stress = \frac{{{ω _{max}} + {ω _{min}}}}{2}\)

\({\sigma _v} = variable\;stress = \frac{{{ω _{max}} - {ω _{min}}}}{2}\)

Soderberg’s formula,

\(\frac{1}{{FOS}} = \frac{{{\sigma _m}}}{{{\sigma _y}}} + \frac{{{\sigma _y} \times {k_f}}}{{{\sigma _{ea}} \times {k_{surface}} \times {k_{sz}}}}\)

Calculation:

ω_max = +200 kN, ω_min = -200 kN

ω_m = 0,

Variable load (ω_v) = (ω_max - ω_min)/2 = (200 - (- 200))/2 = 200 kN

Now,

\({\rm{Stress\;ratio\;}} = \frac{{{\sigma _{min}}}}{{{\sigma _{max}}}} = \frac{{{ω _{min}}}}{{{ω _{max}}}} = - 1\)

σ_ea = 1080 × 0.5 K_a

∴ σ_ea = 378 MPa

Now,

\(\frac{1}{{FOS}} = 0 + \frac{{200 \times {{10}^3} \times 1}}{{{d^2}\frac{\pi }{4} \times 378 \times 0.8 \times 0.83}}\)

∴ d = 45.045 mm

Concept:

For Soderberg criterion of design yield strength is considered.

For Goodman’s criterion of design Ultimate tensile strength is considered.

Calculation:

Given:

D = 30 mm, P = 40 KN (compression), P = 160 KN (Tension)

S_yt = 420 MPa, S_ut = 600 MPa, S_en = 240 MPa

Now,

\({\sigma _m} = \;\frac{{\left( { - 40 + 160} \right) \times {{10}^3}}}{{2A}}\;MPa\; = \frac{{60 \times {{10}^3}}}{A}MPa\)

\({\sigma _v} = \;\frac{{\left( {160 - \left( { - 40} \right)} \right) \times {{10}^3}}}{{2A}}\;MPa = \;\frac{{100 \times {{10}^3}}}{A}MPa\)

Here,

\(A = \frac{{\pi {D^2}}}{4} = \;\frac{{\pi {{30}^2}}}{4}\;m{m^2} = 706.858\;m{m^2}\)

Now,

According to Soderberg criterion

\(\frac{{{\sigma _m}}}{{{S_{yt}}}} + \;\frac{{{\sigma _v}}}{{{S_{en}}}} = \;\frac{1}{N}\;\)     ---(1)

Substituting all the values in (1), we get

∴ N = 1.26 (option a)

Now,

According to Goodmann’s criterion

\(\frac{{{\sigma _m}}}{{{S_{ut}}}} + \frac{{{\sigma _v}}}{{{S_{en}}}} = \frac{1}{N}\)   ---(2)

Substituting all the values in (2), we get

∴ N = 1.37 (option b)

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: