Past Year Questions: Bending & Shear Stresses | Solid Mechanics - Mechanical Engineering PDF Download

Q1: The three-dimensional state of stress at a point is given by   [2024, Set-l]

The maximum shear stress at the point is 
(a) 25MPa 
(b) 5MPa 
(c) 15MPa 
(d) 20MPa
Ans: (d)
In the given matrix, shear stresses are zero. So, the elements on diagonal of this matrix are principal stress. So,

Q2: A 2D thin plate with modulus of elasticity, E = 1.0 N/m² , and Poisson's ratio, μ = 0.5, is in plane stress condition. The displacement field in the plate is given by μ = C²y and v = 0, where us and v are displacements (in m) along the X and Y directions, respectively, and C is constant (in m⁻²). The distance x and y along X an Y, respectively, are in m. The stress in the X direction is σ _xx  =40xy N/m² , and the shear stress is τ_xy = ax² N/m². What is the value of α (in N/m⁴ , in integer) ? [2023, Set-lI]
(a) 30
(b) 40
(c) 45
(d) 55
Ans: (b)





Q3: The components of pure shear strain in a sheared material are given in the matrix form:  [2022, Set-lI]

Here, Trace(ε) = 0. Given, P = Trace(ε⁸) and Q = Trace(ε¹¹). 
The numerical value of (P + Q) is ________. (in integer)
(a) 12
(b) 28
(c) 32
(d) 46
Ans: (c)
∣A − λI∣ = 0


λ = ±√2
Eigen values of ε are √2  and − √2  
Eigen values of ε⁸ are (√2)⁸ and (−√2)⁸
Eigen values of ε¹¹ are (√2)¹¹ and (−√2)¹¹
P = Trace(ε⁸) = sum of Eigen values = (√2)⁸ + (−√2)⁸ = 32  
Q = Trace(ε¹¹) = sum of Eigen values = (√2)¹¹ + (−√2)¹¹  
P + Q = 32 + 0 = 32

Q4: The hoop stress at a point on the surface of a thin cylindrical pressure vessel is computed to be 30.0 MPa. The value of maximum shear stress at this point is  [2022, Set-I]
(a) 7.5 MPa
(b) 15.0 MPa
(c) 30.0 MPa
(d) 22.5 MPa
Ans: (a)
Given,
Hoop stress (σ_h) = pd/2t = 30 MPa
Maximum shear stress in plane (τ_max) in plane= 7.5 Mpa

Q5: For a channel section subjected to a downward vertical shear force at its centroid, which one of the following represents the correct distribution of shear stress in flange and web?  [2019 : 1 Mark, Set-ll]
(a)

(b)

(c)

(d)

Ans: (c)
Shear flow distribution for channel.

Q6: Cross section of a built-up wooden beam as shown in figure (not drawn to scale) is subjected to a vertical shear force of 8 kN. The beam is symmetrical about the neutral axis (NA), shown, and the moment of inertia about N.A. is 1.5 x 10⁹ mm⁴. Considering that the nails at the location P are spaced longitudinally (along the length of the beam) at 60 mm, each of the nails at P will be subjected to the shear force of [2019 : 2 Marks, Set-I]
(a) 240 N
(b) 480 N
(c) 60 N
(d) 120 N
Ans: (a)
Shear Flow,

Distance between two nails l = 60 mm
∴ S.F. resisted by each nail = q x l = 240 N

Q7: For a given loading on a rectangular plain concrete beam with an overall depth of 500 mm, the compressive strain and tensile strain developed at the extreme fibers are of the same magnitude of 2.5 x 10^-4. The curvature in the beam cross-section (in m^-1, round off to 3 decimal places), is _______. [2019 : 1 Mark, Set-I]
Ans:
Given: D= 500 mm


⇒
⇒

= 1*10^-6

Q8: An 8 m long simply-supported elastic beam of rectangular cross-section (100 mm x 200 mm) is subjected to a uniformly distributed load of 10kN/m over its entire span. The maximum principal stress (in MPa, up to two decimal places) at a point located at the extreme compression edge of a cross-section and at 2 m from the support is ______ .    [2018 : 2 Marks, Set-II]
Ans:

[Due to symmetry]
M_A = (-10 x 2 x 1) + 40 x 2
= 60 kNm
σ = My/I

Direct shear stress = 0
Principal stress,

So principal stress
= 90 N/mm² = 90 MPa

Q9: A cantilever beam of length 2 m with a square section of side length 0.1 m is loaded vertically at the free end. The vertical displacement at the free end is 5 mm. The beam is made of steel with Young’s modulus of 2.0 x 10¹¹ N/m². The maximum bending stress at the fixed end of the cantilever is [2018 : 2 Marks, Set-I]
(a) 20.0 MPa
(b) 37.5 MPa
(c) 60.0 MPa
(d) 75.0 MPa
Ans: (b)


⇒
= 37.5 x 10⁶ N/m² = 37.5 MPa

Q10: A 450 mm long plain concrete prism is subjected to the concentrated vertical loads as shown in the figure. Cross section of the prism is given as 150 mm x 150 mm. Considering linear stress distribution across the cross-section, the modulus of rupture (expressed in MPa) is ___ .   [2016 : 2 Marks, Set-II]
Ans:
BM_Q = 11.25 x 150
= 1.6875 x 10⁶ N-mm
⇒
where,
⇒ 

Q11: A simply supported reinforced concrete beam of length 10 m sags while undergoing shrinkage. Assuming a uniform curvature of 0.004 m-¹ along the span, the maximum deflection (in m) of the beam at mid-span is ______. [2015 : 2 Marks, Set-II]
Ans: 
Method - I
Radius,

= 249.95 m
Deflection = AA' = 250 - 249.95
= 0.05 m
Method-ll

1/R = 0.004

⇒

Q14: Consider a simply supported beam with a uniformly distributed load having a neutral axis (NA) as shown. For points P(on the neutral axis) and Q (at the bottom of the beam) the state of stress is best represented by which of the following pairs?    [2011 : 1 Mark]




Ans: