GATE Past Year Questions: Static, Dynamic Loading & Failure Theories | Design of Machine Elements - Mechanical Engineering PDF Download

Q1: Which one of the following failure theories is the most conservative design approach against fatigue failure?  [GATE ME 2024]
(a) Soderberg line
(b) Modified Goodman line
(c) Gerber line
(d) Yield line
Ans: (a)
Gerber Line. A parabolic curve joining S_e on the ordinate to S_ut on the abscissa is called the Gerber line. 
Soderberg Line A straight line joining S_e on the ordinate to S_yt  on the abscissas called the Soderberg line. 
Goodman Line. A straight line joining S_e on the ordinate to  S_ut  on the abscissa is called the Goodman line.

The Gerber parabola fits the failure points of test data in the best possible way. The goodman line fits beneath the scatter of this data. Both Gerber parabola and Goodman line intersect at ( S_e ) on the ordinate to ( S_ut ) on the abscissa. However, the Goodman line is more safe from design considerations because it is completely inside the Gerber parabola and inside the failure points. The Soderberg line is a more conservative failure criterion and there is no need to consider even yielding in this case.

Q1: A structural member under loading has a uniform state of plane stress which in usual notations is given by σ _x = 3 P, σ _y = − 2 P , τ _{x y} = √2 P where P > 0 . The yield strength of the material is 350 MPa. If the member is designed using the maximum distortion energy theory, then the value of P P at which yielding starts (according to the maximum distortion energy theory) is [GATE ME 2022,SET-2]
(a) 70 Mpa
(b) 90 Mpa
(c) 120 Mpa
(d) 75 Mpa
Ans: (a)
σ _x = 3 
σ _y = − 2 P
τ = √2 P 
According to maximum distortion energy theory,

P x 5 = 350
P = 70 Mpa

Q1: The von Mises stress at a point in a body subjected to forces is proportional to the square root of the  [GATE ME 2021 SET-2]
(a) total strain energy per unit volume
(b) plastic strain energy per unit volume
(c) dilatational strain energy per unit volume
(d) distortional strain energy per unit volume
Ans: (d) 
Condition for failure as per M.D.E.T.
Distortion energy per unit volume under tri-axial state of stress > Distortion energy per unit volume under uni-axial state of stress.

 Hence,

   (Von Mises effective stress)

S_yt = Von Mises effective stress is defined as the uni-axial yield stress that would create same distortion energy created by the tri-axial state of stress.

Q2: A machine part in the form of cantilever beam is subjected to fluctuating load as shown in the figure. The load varies from 800 N to 1600 N. The modified endurance, yield and ultimate strengths of the material are 200 MPa, 500 MPa and 600 MPa, respectively.  [GATE ME 2021, SET-1]

The factor of safety of the beam using modified Goodman criterion is _______ (round off to one decimal place).
Ans: (1.9 to 2.1)
A : Critical Point
σ_{max, A} = σ_{b, max} at A due to 1600 N = 6M/bd² = 6 x 1600 x 200/12 x (2σ)²
σ _max = 200MPa
σ_{min, A} = σ_{b, max} at A due to 800 N
= 6 x 800 x 200/12 x (20)2 = 100MPa

150/600 + 50/200 ≤ 1/N
N ≤ 2
N ≈ 2
Langer Criterion:
≤ 1/N
150/500 + 50/500 ≤ 1/N (N ≤ 2.5)
N = 2.5
Modified Goodman = Safe result of [Goodman or Langer]

Question for GATE Past Year Questions: Static, Dynamic Loading & Failure Theories

Try yourself:Consider the two states of stress as shown in configurations I and II in the figure below.
From the standpoint of distortion energy (vonMises) criterion, which one of the following statements is true?

[2014, Set-2]

• A.

  I yields after II
• B.

  II yields after I
• C.

  Both yield simultaneously
• D.

  Nothing can be said about their relative yielding

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Question for GATE Past Year Questions: Static, Dynamic Loading & Failure Theories

Try yourself:Consider the schematic of a riveted lap Joint subjected to tensile load F, as shown below.
Let d be the diameter of the rivets, and S_f be the maximum permissible tensile stress in the plates. What should be the minimum value for the thickness of the plates to guard against tensile failure of the plates? Assume the plates to be identical.

[2017, Set-1]

• A.

  
• B.

  
• C.

  
• D.

  

Explanation

Tensile stress

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Question for GATE Past Year Questions: Static, Dynamic Loading & Failure Theories

Try yourself:For the given fluctuating fatigue load, the values of stress amplitude and stress ratio are respectively

[2015, Set-3]

• A.

  100MPa and 5
• B.

  250 MPa and 5
• C.

  100 MPa and 0.20
• D.

  250 MPa and 0.20

Explanation

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Question for GATE Past Year Questions: Static, Dynamic Loading & Failure Theories

Try yourself:A carpenter glues a pair of cylindrical wooden logs by bonding their end faces at an angle of θ = 30° as shown in the figure.

The glue used at the interface fails if
Criterion 1 : the maximum normal stress exceeds 2.5 MPa
Criterion 2 : the maximum shear stress exceeds 1.5 MPa
Assume that the interface fails before the logs fail. When a uniform tensile stress of 4 MPa is applied, the interface

[2018, Set-1]

• A.

  fails only because of criterion 1
• B.

  fails only because of criterion 2
• C.

  fails because of both criteria 1 and 2
• D.

  does not fail.

Explanation

Normal stress on inclined plane
σ = σ_x cos²θ
= 4 × cos² 30 = 3 MPa
Shear stress on inclined plane

= 2 × sin 60° = 1.73 MPa
Since both the stress exceeds the given limits, answer is option (c).

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Question for GATE Past Year Questions: Static, Dynamic Loading & Failure Theories

Try yourself:A large uniform plate containing a rivet hole subjected to uniform uniaxial tension of 95 MPa.
The maximum stress in the plate is

[1992]

• A.

  100 MPa
• B.

  285 MPa
• C.

  190 MPa
• D.

  Indeterminate

Explanation

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