Deflection, Energy Methods and Buckling - Mechanics of Deformable Bodies - Engineering Sciences Previous Year Questions

1

2012 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2012

For the beam-column configurations shown in figure, the minimum Euler buckling load is obtained for the case (Young’s modulus and second moment of cross-sectional area are as indicated)

A

(i)

B

(ii)

C

(iii)

D

(iv)

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2

2012 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2012

Consider a simply supported beam loaded either by a uniformly distributed transverse load or by a concentrated transverse load applied at the center such that the maximum bending stress in both cases is the same. The ratio of the strain energy for the two cases is

A

4/5

B

5/8

C

8/5

D

1

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3

2012 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2012

The vertical deflection at the ends is

A

4.92 mm

B

4.58 mm

C

9.31 mm

D

9.84 mm

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4

2012 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2012

The value of the additional end moment M (in N.m) required to obtain an upward deflection of 1 mm at the free end, is (moment is positive in counterclockwise direction)

A

-533.33

B

533.33

C

-528

D

528

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5

2012 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2012

A small railway bridge is constructed from identical steel truss members, each of length \(l\), cross-sectional area \(A\) and Young’s modulus \(E\). A train stops on the bridge. The loads applied by the train on the truss on one side of the bridge may be assumed to act at pins A, B and C, as shown. The displacement of the support C due to this loading is

A

0

B

\(Ql/(AE)\)

C

\(\sqrt{3}Ql/(4AE)\)

D

\(Ql/(\sqrt{3}AE)\)

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6

2012 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2012

If the deflection at the free-end (under load P, with end moment M=0) is measured as δ = 5 mm, the flexural rigidity EI for the beam is (in N m²)

A

66,666

B

66,000

C

67,300

D

64,000

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7

2013 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2013

A rigid bar AB is hinged at B through a torsional spring with spring constant \(k_t\). For small rotations of the bar AB about B, the critical load \(P_{cr}\) is given by

A

\(\frac{k_t}{2L}\)

B

\(\frac{k_t}{L}\)

C

\(\frac{2 k_t}{L}\)

D

\(\frac{4 k_t}{L}\)

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8

2013 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2013

A beam is fixed at the left end and supported by a spring at the other end. The length of the beam is L and its flexural rigidity is EI. The spring constant of the spring is \(k = \frac{3 EI}{L^3}\). A vertical downward load P is applied at the right end. The deflection of the point under the load P is

A

\(\frac{P L^3}{9 EI}\)

B

\(\frac{P L^3}{6 EI}\)

C

\(\frac{2 P L^3}{9 EI}\)

D

\(\frac{5 P L^3}{9 EI}\)

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9

2013 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2013

Find the maximum bending moment (magnitude wise) in kN-m for the beam shown in the Figure.

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10

2013 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2013

Neglecting the axial compression of member AB, the deflection of point C in the direction of the load is

A

\(\frac{2 P L^3}{E a^4}\)

B

\(\frac{4 P L^3}{E a^4}\)

C

\(\frac{8 P L^3}{E a^4}\)

D

\(\frac{16 P L^3}{E a^4}\)

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11

2013 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2013

A disk of mass \(M= 14\text{kg}\) and radius \(1 \text{ m}\) is attached to a spring which has a stiffness \(k = 75 \text{ N/m}\) and an unstretched length of \(1\text{ m}\). If the disk is released from rest in the position shown in the Figure and the disk rolls without slipping, find its angular velocity (in rad/s) at the instant the center of mass is displaced by \(3 \text{ m}\).

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12

2014 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2014

The Euler’s buckling load of a column fixed at both the ends is P. If one of the ends is made free, the buckling load shall change to

A

P/16

B

P/8

C

P/4

D

P/2

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13

2014 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2014

The supporting structure of a water tank is made of reinforced concrete (RC) with a tubular cross section of inner diameter $d_i$, outer diameter $d_o$, height $l$, and Young's modulus $E$. The mass of the tank is $m$. If mass of the supporting structure is neglected, then the natural frequency of the water tank in transverse direction is

A

$\sqrt{\frac{3\pi E(d_o^4 - d_i^4)}{64 l^3 m}}$

B

$\sqrt{\frac{\pi E(d_o^4 - d_i^4)}{8 l^3 m}}$

C

$\sqrt{\frac{384 \pi E(d_o^4 - d_i^4)}{360 l^3 m}}$

D

$\sqrt{\frac{\pi E(d_o^4 - d_i^4)}{64 l^3 m}}$

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14

2014 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2014

The vertical deflection at the free end of the cantilever beam as shown in figure is
\(EI =\) flexural rigidity

A

\(1400/EI\)

B

\(1400/3EI\)

C

\(200/EI\)

D

\(100/EI\)

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15

2014 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2014

A cantilever beam is subjected to following three different loading conditions:
(a) a concentrated load \( P \) at its free end,
(b) a couple \( M_o \) at its free end and
(c) both loads acting simultaneously
The flexural rigidity of the beam may be assumed as \( EI \). The strain energy due to bending when both loads act simultaneously

A

can be determined by applying the principle of superposition and the strain energy is \(\frac{P^2 L^3}{6EI} + \frac{M_o^2 L}{2EI} \)

B

can be determined by applying the principle of superposition and the strain energy is \(\frac{P^2 L^3}{6EI} + \frac{M_o^2 L}{2EI} \)

C

cannot be determined by applying the principle of superposition and the strain energy is \(\frac{P^2 L^3}{6EI} + \frac{M_o^2 L}{2EI} + \frac{P M_o L^2}{2EI} \)

D

cannot be determined by applying the principle of superposition and the strain energy is \(\frac{P^2 L^3}{6EI} + \frac{M_o^2 L}{2EI} + \frac{P M_o L^2}{2EI} \)

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16

2015 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2015

The plane frame shown is analyzed by neglecting axial and shear deformations. The horizontal displacement of joint B is

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17

2015 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2015

Two massless rigid bars, each of length \(a = 0.5m\), are connected by a rotational spring having stiffness \(k = 1000 N.m/rad\). Find the buckling load \(P\) (in kN).

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18

2016 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2016

For a slender steel column of circular cross-section the critical buckling load is P_cr. If the diameter of the column is doubled (keeping other material and geometrical parameters same), then the critical buckling load of the column is

A

P_cr /16

B

8 P_cr

C

2 P_cr

D

16 P_cr

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19

2016 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2016

A beam having flexural rigidity \(EI\) and length \(L\) is subjected to a concentrated end moment \(M_0\) as shown in the figure. For \(EI=4 \times 10^7~N\cdot m^2\), \(L=1~m\) and \(M_0=8~kN\cdot m\), the strain energy stored (in kN-m) in the beam and the rotation (in rad) at the free end respectively are

A

8.00 and 0.02

B

8.00 and 2.00

C

8.00 and 0.04

D

0.80 and 2.00

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20

2016 · Engineering Sciences · Mechanics of Deformable Bodies · Deflection, Energy Methods and Buckling

Engineering Sciences (XE) 2016

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