Firstranker's choice
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GATE 2015
D : SOLID MECHANICS
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Q. 1 – Q. 9 carry one mark each.
Q.1 Find the force (in kN) in the member BH of the truss shown.
C
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5m
B 2m D
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3m I2m 2m
JO
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A G E
4m H 2m
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2m F 4m40 kN 40 kN 40 kN
Q.2 Consider the forces of magnitude F acting on the sides of the regular hexagon having side length a.
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At point B, the equivalent force and couple are, respectively,
B
F
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F
F
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Fa
F
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(A) F(?) and 3v3Fa (clockwise)(B) F(?) and v3Fa (clockwise)
(C) F(?) and v3Fa (counter clockwise)
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(D) F(?) and 3v3Fa (counter clockwise)
F
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Q.3 Bar-1 has a diameter d, length L, and elastic modulus E and subjected to tensile load P, resulting in
an elongation of ?1. Bar-2 has diameter, 2d, length 2L, an elastic modulus 2E and subjected to
tensile load 2P, resulting in an elongation of ?2. Find the ratio ?1/?2.
Q.4 In a plane stress problem, the principal stresses at a point are 30MPa and -15MPa. At the same
point, on an element whose sides make an angle of 45° with respect to the principal axes, the
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normal stresses (in MPa) are(A) 15/2 and 15/2 (B) 30/2 and 30/2 (C) 15/2 and -15/2 (D) 30/2 and -30/2
Q.5 Two systems shown below start from rest. For the system shown on the left, two 50N blocks are
connected by a cord. For the system shown on the right, the 50N block is pulled by a 50N
downward force. Neglect friction. Which of the following is true?
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A
50 N
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50 N BC
50 N
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50 N
Q.6 (A) Blocks A and C have the same acceleration.
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(B) Block C will have a larger acceleration than block A.(C) Block A will have a larger acceleration than block C.
(D) Block A will not move.
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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).
P
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Ca
k
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B
a
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AQ.7 A simply supported beam having a rectangular cross-section of depth d is subjected to a vertical
concentrated load P at the mid-span. The maximum shear stress in a section occurs at
(A) d/2 from the top of the cross-section
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(B) d/3 from the top of the cross-section
(C) 2d/3 from the top of the cross-section
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(D) Top of the cross-sectionQ.8 A steel block of size 100×50×25 mm³ is subjected to a uniform pressure on all faces. The
dimension of the 100mm edge reduces by 25µm (note 1 µm = 10¯ºm). Find the applied pressure (in
GPa). Use E = 240 GPa and ? = 0.3.
Q.9 Which one of the following statements is true?
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(A) In a tensile test on a rod made of ductile material, failure occurs along a plane making 45° with
respect to the axis of the rod
(B) In a tensile test on a rod made of brittle material, failure occurs along a plane making 45° with
respect to the axis of the rod
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(C) In a torsion test on a rod made of ductile material, failure occurs along a plane making 45° with
respect to the axis of the rod
(D) In a torsion test on a rod made of brittle material, failure occurs along a plane making 0° with
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respect to the axis of the rodQ. 10 – Q. 22 carry two marks each.
Q.10 A block is travelling with a constant speed v0 on a smooth surface when the surface suddenly
becomes rough with a coefficient of friction µ, which causes the block to stop after a distance d.
When the block travels twice as fast, i.e. at a speed 2v0, it travels a distance D on the rough surface
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before stopping. Find the ratio D/d.
V0
d
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Smooth Rough
Q.11 The beam shown below is loaded with a concentrated clockwise moment of 80kN-m at point B.
The bending moment diagram (in kN-m) is
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20 kN
A 4m B 4m C 4m D
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80 kN-m El constant
(A)
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80C D
A B
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(B)
80
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40
B C D
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A
40
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80(C) A B C D
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80(D) W B C D
80
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80
Q.12 The beam shown has an internal hinge at B. A vertical load P = 25kN is applied at B. Use L = 2m.
Magnitude of the reactions (i.e. forces and moments) at A and C are
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P
L
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LC
A
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B
777
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(A) Vertical reaction force at C is 12.5kN, vertical reaction force at A is 12.5kN, moment reaction
at A is 0kN-m.
(B) Vertical reaction force at C is 0kN, vertical reaction force at A is 25kN, moment reaction at A
is 50kN-m.
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(C) Vertical reaction force at C is 25kN, vertical reaction force at A is 0kN, moment reaction at A
is 50kN-m.
(D) Vertical reaction force at C is 0kN, vertical reaction force at A is 25kN, moment reaction at A
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is 25kN-m.Q.13 Blocks P and Q are released from rest in the positions shown. Neglect friction between all surfaces,
i.e., both blocks can translate freely. Then the direction of the acceleration of block Q (i.e. aQ) is
Q
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?
P
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(A)
(C)
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AQ (B)?
ao
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AQ
(D)
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?aq
Q.14 The acceleration, a, of a particle as a function of its position, x, is given by the relation
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a = 0. 1 + sin(x/b), where a and x are expressed in m/s² and meters, respectively. Consider b = 1m.When x = 0, velocity is v = 1m/s. Find v (in m/s) when x = p meters.
Q.15 The 30kg block B shown below is suspended by a 2m cord attached to the 60kg cart A. Friction is
negligible. If the system is released from rest in the position shown, find the ratio of the velocity
magnitudes
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|VA|/|VB| when the cord is vertical.
A
60 kg
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Cord
B
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25°30 kg
Q.16 The plane frame shown has an internal hinge at C. Find the magnitude of axial force (in kN) in
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member BC.
200 kN
C
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D
B
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4mA
3m 3m E
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F
3m
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Q.17 Two 50mm diameter solid steel rods are rigidly connected together at right angles and loaded as
shown. Use P = 1000p kN. At point A, located at the top of the cross-section at the fixed end, the
magnitude of bending stress (s) and shear stress (t) are
W
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Z
2 m
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?A
(?) s = 256 MPa, t = 512 MPa
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(C) s = 512 MPa, t = 128 MPa
P
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1 mX
(?) s = 512 MPa, t = 256 MPa
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(D) s = 128 MPa, t = 512 MPa
Q.18 At a temperature of 40°C, a rod tightly fits between two rigid walls such that the compressive stress
in the rod is 60MPa. Given E = 200GPa and a = 20×10-6/°C, find the temperature at which the rod
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will just lose contact with the walls.Q.19 A massless rigid rod AB of length h is pinned at end A and carries mass m at end B. The rod is also
supported by two linear springs of stiffness k at a height d from the end A. Use m = 4kg, h = 0.5m,
d = 0.2m, k = 600N/m and g = 10 m/s². For small oscillations about the position shown, find the
frequency of free vibration (in rad/s).
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g
Bm
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hk
k
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d
A
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Q.20 Find the maximum force P (in kN) that can be applied to the planar structure ABC so as to prevent
buckling in any of the members. Consider buckling only in the plane of the structure. Joint B is a
pin connection. Use E = 200GPa for both members. The diameter of member AB is 10 mm and the
diameter of member BC is 15 mm.
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P? = 45°
1m
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?
A B
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2mAB = 10 mm diameter
BC = 15 mm diameter
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C
Q.21 The plane frame shown is analyzed by neglecting axial and shear deformations. The horizontal
displacement of joint B is
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C
L
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PB
L
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A
(A) 2PL³/3EI (B) PL³/EI (C) 3PL³/2EI (D) PL³/2EI
Q.22 A thin walled cylindrical pressure vessel having mean radius 100mm and wall thickness 5mm, is
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subjected to internal pressure p. If the factor of safety is 2 and the yield stress in shear is 100MPa,find the maximum value of p (in MPa).
Pressure p
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200mmEND OF THE QUESTION PAPER
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