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GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering PDF Download

Q1: The figure shows a thin cylinder pressure vessel constructed by welding plates together along a line that makes an angle α = 60º with the horizontal. The closed vessel has a wall thickness of 10 mm and diameter of 2m. When subjected to an internal pressure of 200kPa, the magnitude of the normal stress acting on the weld is ______ MPa (rounded off to 1 decimal place).   (2024)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(a) 12.5
(b) 5.5
(c) 18.6
(d) 22.6
Ans:
(a)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q2: Ignoring the small elastic region, the true stress (σ) − true strain (ε) variation of a material beyond yielding follows the equation σ =400ε0.3 MPa. The engineering ultimate tensile strength value of this material is ________ MPa. (Rounded off to one decimal place)     (2024)
(a) 145.6
(b) 325.6
(c) 206.6
(d) 125.2
Ans: 
(c)
Sol:

GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

The true strain (εT) at the onset of necking equal to the strain hardening exponent i.e. at ultimate tensile point.
So, 𝜀T=n=0.3 at ultimate point
So, from equation (i)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q3: The principal stresses at a point P in a solid are 70 MPa, -70 MPa and 0. The yield stress of the material is 100 MPa. Which prediction(s) about material failure at P is/are CORRECT?     (2024)
(a) Maximum normal stress theory predicts that the material fails
(b) Maximum shear stress theory predicts that the material fails
(c) Maximum normal stress theory predicts that the material does not fail
(d) Maximum shear stress theory predicts that the material does not fail
Ans:
(b, c)
Sol: 
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
For maximum shear stress theory:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
So material will fail As per maximum normal stress theory:

𝜎1 and sigma2 > sigmayt 
then material will fail
Here 70 & -70 < 100
So material is safe.

Q4: A prismatic bar PQRST is subjected to axial loads as shown in the figure. The segments having maximum and minimum axial stresses, respectively, are    (2024)

GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

(a) QR and PQ
(b) ST and PQ
(c) QR and RS
(d) ST and RS
Ans:
(d)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
Hence, maximum and minimum axial stresses are in ST and RS portions because of prismatic bar.

Q5: The loading and unloading response of a metal is shown in the figure. The elastic and plastic strains corresponding to 200 MPa stress, respectively, are   (2021 Set- 1)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

(a) 0.01 and 0.01
(b) 0.02 and 0.01
(c) 0.01 and 0.02
(d) 0.02 and 0.02
Ans:
(b)
Sol: Elastic strain : Which can be recovered = 0.03 - 0.01 = 0.02
Plastic strain : Permanent strain = 0.01

Q6: Uniaxial compression test data for a solid metal bar of length 1 m is shown in the figure.   (2020 Set- 2)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

The bar material has a linear elastic response from O to P followed by a non-linear response. The point P represents the yield point of the material. The rod is pinned at both the ends. The minimum diameter of the bar so that it does not buckle under axial loading before reaching the yield point is _______ mm (round off to one decimal place).
(a) 16.56
(b) 56.94
(c) 78.25
(d) 68.42
Ans:
(b)
Sol:

GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q7: Bars of square and circular cross-section with 0.5 m length are made of a material with shear strength of 20 MPa. The square bar cross-section dimension is 4 cm x 4 cm and the cylindrical bar cross-section diameter is 4 cm. The specimens are loaded as shown in the figure.   (2020 Set- I)

GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Which specimen(s) will fail due to the applied load as per maximum shear stress theory?
(a) Tensile and compressive load specimens
(b) Torsional load specimen
(c) Bending load specimen
(d) None of the specimens
Ans:
(a)
Sol: 
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q8: Consider a linear elastic rectangular thin sheet of metal, subjected to uniform uniaxial tensile stress of 100 MPa along the length direction. Assume plane stress conditions in the plane normal to the thickness. The Young's modulus E=200 MPa and Poisson's ratio v=0.3 are given. The principal strains in the plane of the sheet are   (2019 Set -2)
(a) (0.35, -0.15)
(b) (0.5, 0.0)
(c) (0.5, -0.15)
(d) (0.5,-0.5)
Ans:
(c)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q9:  In a UTM experiment, a sample of length 100 mm, was loaded in tension until failure. The failure load was 40 kN. The displacement, measured using the cross-head motion, at failure, was 15 mm. The compliance of the UTM is constant and is given by 5 x 10-8  m/N. The strain at failure in the sample is _______%.    (2019 Set -1)
(a) 8
(b) 16
(c) 13
(d)  6

Ans: (c)
Sol: 
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q10: A plane-strain compression (forging) of a block is shown in the figure. The strain in the z-direction is zero. The yield strength (Sy) in uniaxial tension/compression of the material of the block is 300 MPa and it follows the Tresca (maximum shear stress) criterion. Assume that the entire block has started yielding. At a point where σx = 40M Pa, (compressive) and  τxy = 0, the stress component σy  is  (2019 Set - 1)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

(a) 340 MPa (compressive)
(b) 340 MPa (tensile)
(c) 260 MPa (compressive)
(d) 260 MPa (tensile)
Ans: 
(a)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q11: Consider the stress-strain curve for an ideal elastic-plastic strain hardening metal as shown in the figure. The metal was loaded in uniaxial tension starting from O. Upon loading, the stress-strain curve passes through initial yield point at P, and then strain hardens to point Q, where the loading was stopped. From point Q, the specimen was unloaded to point R, where the stress is zero. If the same specimen is reloaded in tension from point R, the value of stress at which the material yields again is _________ MPa.    (2019 Set-1)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering(a) 210
(b) 70
(c) 420
(d) 480
Ans: 
(a)
Sol: Yield strength will increase to 210 MPa due to strain hardening.
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q12: A solid cube of side 1 m is kept at a room temperature of 32ºC. The coefficient of linear thermal expansion of the cube material is 1 x 10-5/ºC and the bulk modulus is 200 GPa.If the cube is constrained all around and heated uniformly to 42ºC, then the magnitude of volumetric (mean) stress (in MPa) induced due to heating is____   (2019 Set-1)
(a) 20
(b)60
(c) 80
(d) 100
Ans:
(b)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q13: The State of stress at a point, for a body in place stress, is shown in the figure below. If the minimum principal stress is 10 kPa, then the normal stress σs (in kPa) is    (2018)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(a) 9.45
(b) 18.88
(c) 37.78
(d) 75.50
Ans: 
(c)
Sol:
σx = 100 k Pa, txy = 50 k Pa
Minimum principal stress
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringBy squaring
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q14: The state of stress at a point on an element is shown in figure (a). The same state of stress is shown in another coordinate system in figure (b).   (2016)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringThe components (txx, tyy, txy,) are given by 

(a)  GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

(b) (0, 0, p) 

(c)GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

(d) GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
Ans: 
(b)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringHere θ = - 45
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering = 0
When θ = +45
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q15: The principal stresses at a point inside a solid object are σ1 = 100 MPa, σ2 = 100 MPa and σ3 = 0 MPa. The yield strength of the material is 200 MPa. The factor of safety calculated using Tresca (maximum shear stress) theory is nT and the factor of safety calculated using Von Mises(maximum distortional energy) theory is nv. Which one of the following relations is TRUE?    (2016)

(a) GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(b) GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(c) nT = nv
(d) GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Ans: (c)
Sol: 

GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering→ Von mises theoryGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering∴ nT = nv

Q16: The principal stresses at a point in a critical -section of a machine component are σ1 = 60 MPa, σ2 = 5 MPa and σ3 = - 40 MPa. For the material of the component, the tensile yield strength is σy = 200 MPa. According to the  maximum shear stress theory, the factor of safety is     (2016)

(a) 1.67
(b) 2.00
(c) 3.60
(d) 4.00
Ans: 
(b)
Sol:
σ1 = 60 MPa, σ2 = 5MPa σ3 = - 40 MPa
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q17: In a plane stress condition, the components of stress at a point are σx = 20 MPa, σy = 80 MPa and σxy = 40 MPa. The maximum shear stress (in MPa) at the point is
(a) 20
(b) 25
(c) 50
(d)  100
Ans: (c)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

 Q18: The homogenous state of stress for a metal part undergoing plastic deformation is
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringWhere the stress component values are in MPa. Using Von Mises yield criterion, the value of estimated shear yield stress, in MPa is    (2012)

(a) 9.50
(b) 16.07
(c) 28.52
(d) 49.41
Ans: (b)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringWe know, σ11 = 10, σ22 = 20,
σ33 = -10;
σ12 = 5;
σ23 = σ13 = 0
∴ σeq = 27.839 MPa
Shear stress at yield, 

GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q19: Match the following criteria of material failure, under biaxial stresses σ1 and σ2 and yield stress σy, with their corresponding graphic representations:    (2011)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering 
(a)  P–M, Q–L, R–N
(b) P–M, Q–N, R–L
(c) P–N, Q–M, R–L
(d) Q–N, Q–L, R–M
Ans:
(c)

Q20: The state of plane-stress at a point is given by σx =200 MPa, σy =100 MPa and τxy = 100 MPa. The maximum shear stress
(a) 111.8
(b) 150.1
(c) 180.3
(d) 223.6
Ans:
(a)
Sol:
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering= 111.80 MPa. 

Q21: If the principal stresses in a plane stress problem are σ1 = 100 MPa, σ2 = 40 MPa, the magnitude of the maximum shear stress (in MPa) will be   (2009)
(a) 60
(b) 50
(c) 30
(d) 20
Ans:
(c)
Sol: 

Maximum shear stress,
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q22: According to Von-Mises' distortion energy theory, the distortion energy under three dimensional stress state is represented by    (2006)
(a) GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(b) GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(c) GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(d) GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
Ans: (c)
Sol: 
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

 

 Q23: A shaft subjected to torsion experiences a pure shear stress x on the surface. The maximum principal stress on the surface which is at 45° to the axis will have a value   (2003)
(a) τ cos 45° 
(b) 2τ cos 45°
(c) τ cos2 45°
(d) 2τ sin 45° cos 45°
Ans: (d)
Sol: 
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q24: If the two principal strains at a point are 1000 x 10-6 and –600 × 10–6, then the maximum shear strain is    (1996)
(a) 800 x 10-6
(b) 500 x 10-6
(c) 1600 x 10-6
(d) 200 x 10-6
Ans: (c)
Sol: 

GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q25: At a point in a stressed body the state of stress on two planes 45° apart is as shown below. Determine the two principal stresses in MPa.
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(a) 8.242, 0.658
(b) 8.242, 0.758 
(c) 9.234, 0.757
(d) 9.242, 0.758
Ans: (d)
Sol: 

GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
Q26: A large uniform plate containing a rivet hole is subjected to uniform uniaxial tension of 95 MPa. The maximum stress in the plate is    (1992)
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering
(a) 100 MPa
(b) 285 MPa
(c) 190 MPa
(d) Indetermine
Ans: (b)
Sol: 
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q27: The three-dimensional state of stress at a point is given by
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringThe shear stress on the x-face in y-direction at the same point is then equal to     (1990)
(a) zero MN/m2
(b) -10 MN/m2
(c) 10 MN/m2
(d) 20 MN/m2
Ans: (c)
Sol: 
GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical EngineeringGATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering

Q28: An elastic body is subjected to a tensile stress X in a particular direction and a compressive stress Y in its perpendicular direction. X and Y are unequal in magnitude. On the plane of maximum shear stress in the body there will be   (1989)
(a) no normal stress
(b) also the maximum normal stress
(c) the minimum normal stress
(d) both normal stress and shear stress
Ans: (c)

The document GATE Past Year Questions: Principal Stress & Strain | Strength of Materials (SOM) - Mechanical Engineering is a part of the Mechanical Engineering Course Strength of Materials (SOM).
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FAQs on GATE Past Year Questions: Principal Stress & Strain - Strength of Materials (SOM) - Mechanical Engineering

1. What is the difference between principal stress and principal strain?
Ans. Principal stress refers to the maximum and minimum normal stresses experienced by a material in a specific plane, while principal strain refers to the maximum and minimum normal strains experienced by a material in a specific plane.
2. How do you determine the principal stresses in a material?
Ans. The principal stresses in a material can be determined by finding the eigenvalues of the stress tensor matrix, which represent the maximum and minimum normal stresses in a specific plane.
3. What is the significance of principal stresses in mechanical engineering?
Ans. Principal stresses are crucial in mechanical engineering as they help determine the failure criteria of materials and structures, allowing engineers to design components that can withstand varying loading conditions.
4. Can principal stresses be negative?
Ans. Yes, principal stresses can be negative, indicating compressive stresses in a material. It is essential to consider both positive and negative principal stresses when analyzing the structural integrity of a component.
5. How do principal strains affect the deformation behavior of materials?
Ans. Principal strains dictate the direction and magnitude of deformation in a material, influencing its mechanical properties and behavior under load. Understanding principal strains is vital in predicting material failure and designing resilient structures.
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