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GATE 2018 : Engineering Sciences [Section D / Solid Mechanics] (with Final Answer Keys)

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GATE 2018 SOLID MECHANICS (XE-D) XE (D): Q.1 Q. 1 Q. 9 carry one mark each & Q. 10 Q. 22 carry two marks each. A thin cylinder (thickness, and diameter, ) with closed ends is subjected to an internal pressure, . The ratio of hoop stress to longitudinal stress developed in the wall of the cylinder is (A) 1:4 Q.2 (B) 1:2 (C) 1:1 (D) 2:1 A steel rod is fixed at one end and free at the other end. The coefficient of thermal expansion of the steel is , and modulus of elasticity is E. If the temperature of the rod is increased by then the stress and strain developed in the rod are respectively (A) zero, (B) E , (C) E , zero (D) zero, zero Q.3 The effective lengths of the columns with ideal boundary conditions shown in Case-I, Case-II, and Case-III are respectively Hinged-Hinged P Fixed-Free P L Case-I (A) (B) (C) (D) Q.4 Fixed-Fixed P L L Case-II Case-III , , , , / , , , , / At a point in a stressed body, sum of the normal stresses acting on perpendicular faces of an arbitrarily oriented plane stress element is always (A) dependent on the angle of orientation of the element (B) constant and independent of angle of orientation of the element (C) one half of the sum of the principal stresses (D) zero XE-D 1/8 GATE 2018 Q.5 SOLID MECHANICS (XE-D) The principal stresses on a plane stress element are shown in the figure. The maximum shear stress (in MPa) is (A) 200 (B) 100 (C) 50 (D) 0 Q.6 A rigid and thin L-shaped bracket is fixed to the wall at point B, and a force F is applied at point A as shown. For a given force F, the point B experiences the maximum clockwise moment when the inclination (in degrees) with the x-axis is .[up to two decimal places]. Q.7 A cantilever beam with length, L = 1 m, modulus of elasticity, E = 210 GPa, and area moment of inertia, I = . m4 carries a concentrated mass m = 100 kg at its freeend. By idealizing it as a single degree-of-freedom system and neglecting the mass of the cantilever beam, the natural frequency (in rad/s) of small transverse oscillations of the mass m is .. [up to two decimal places] Q.8 For a typical grade of steel, the value of modulus of elasticity (E) and Poisson s ratio ( ) are 208 GPa and 0.3 respectively. The value of shear modulus (G) of the steel (in GPa) is .. XE-D 2/8 GATE 2018 Q.9 SOLID MECHANICS (XE-D) A tensile test is performed on a metallic specimen of diameter 8 mm and gauge length 50 mm. When the tensile load P reaches a value of 20 kN, the distance between the gauge marks increases by 0.09 mm. If the sample remains within the elastic limit, the modulus of elasticity (in GPa) of the test metal is ..[up to two decimal places] P = 20 kN 8 mm 50 mm P = 20 kN Q.10 A beam ABC is subjected to load P at its free end C as shown in the figure. The flexural rigidity of the beam is . The vertical support reaction at point B is / (A) XE-D (B) (C) (D) 3/8 GATE 2018 SOLID MECHANICS (XE-D) Q.11 A horizontal effort P is applied to raise a block of weight W on a rough surface inclined at an angle with the horizontal. If is the coefficient of static friction between the block and the surface, the minimum effort P required to impend the upward motion of the block along the surface is (A) (B) (C) (D) Q.12 Three round bars of same material, equal lengths, and different cross-sectional dimensions are shown in the figures as Case-I, Case-II and Case-III. All the bars are clamped at the upper end, and a concentrated load P is applied at the lower end of each bar. If the elastic strain energy stored in the bar shown in Case-I is then the elastic strain energy stored in Case-II and Case-III respectively is 2d d d L P Case-I Case-II (C) XE-D 8 , , 8 8 L/2 d L/4 P L/4 2d L/2 2d (A) 8 d L/4 L/4 P Case-III (B) (D) 8 , , 8 4/8 GATE 2018 SOLID MECHANICS (XE-D) Q.13 A rigid uniform rod with mass m, length L and center of gravity G is freely suspended from a hinge as shown in the figure. The rod is given a small angular displacement in the counter-clockwise direction from the position in which it hangs vertically ( = 0). If is the acceleration due to gravity, the natural frequency of oscillations (in rad/s) is (B) (A) (D) (C) Q.14 A cylindrical member, made up of ductile material, is subjected to pure torsion as shown in the figure. The failure plane (from Option-I to Option-IV) for ductile material as per maximum shear stress theory is represented by T T Option-I Option-II 45 45 Option-III Option-IV (A) (B) (C) (D) XE-D 90 30 Option I Option II Option III Option IV 5/8 GATE 2018 SOLID MECHANICS (XE-D) Q.15 A simply supported beam of span L is subjected to a couple Mo at a distance a from support A. Among the four options (Option-I to Option-IV) shown, the correct shear force diagram of the beam is Mo A B a L Mo/L (+) Option-I Option-II Mo/L (+) (-) Mo/L Mo a /L (+) Option-III Sign convention of shear force for all options + (-) Mo (a/L-1) Mo/L (+) Option-IV (A) Option-I (B) Option-II (C) Option-III (D) Option-IV Q.16 A circular shaft ABC of diameter, and length, is fixed at end A. It is subjected to the torsional moments at point B and point C as shown in the figure. The ratio of angle of twists at point B to point C is / (A) 1.5 : 1 XE-D (B) 1 : 3 (C) 1 : 2 (D) 1 : 1.5 6/8 GATE 2018 SOLID MECHANICS (XE-D) Q.17 A circular steel bar of diameter 10 mm is bent into the shape as shown in figure, and lies in x-z plane. A horizontal force P is applied along the positive z-direction as shown. The yield strength of the steel is 200 MPa. Neglecting the effect of transverse shear, the load P (in Newton) required to initiate yielding as per maximum shear stress theory of failure is [up to two decimal places] Q.18 Two sliders A and B, connected by a rigid link of length L, slide in two mutually perpendicular and frictionless guide-ways. At a particular instance, the slider A is moving in the downward direction with a speed of 0.05 m/s. At this instance, the magnitude of the velocity of slider B (in m/s) is [up to two decimal places] Q.19 A simply supported beam ABC is subjected to load as shown in the figure. The 20 kN load is applied at point B with the help of a welded bracket as shown. The beam has a rectangular cross-section of 15 mm width and 100 mm depth as shown. The maximum transverse shear stress developed in the beam (in MPa) is ..[up to one decimal place] N . m N/m Enlarged view of the cross-section 100 mm m XE-D . m . m m 15 mm 7/8 GATE 2018 SOLID MECHANICS (XE-D) Q.20 A rigid rod ABC, in the form of a quarter-circular arc of radius R, is hinged at C and supported by a roller at B. A vertical force P is applied at the end A of the bar. For the reactions at B and C to be equal in magnitude, the value of the angle (in degrees) is . [up to two decimal places]. Q.21 A marble of mass m slides along a frictionless linear slide AB kept in a vertical plane. The marble is released from point A with zero initial velocity, and it reaches point B under the action of gravity. Assuming the acceleration due to gravity to be 9.81 m/s2, the speed (in m/s) of the marble when it just reaches point B is [up to two decimal places]. Q.22 A horizontal force 10 N is applied at support B on the frame as shown in figure. Considering only bending deformation, the vertical reaction (in N) at support B is . [up to one decimal place] N END OF THE QUESTION PAPER XE-D 8/8 Q.No. Type Section Key/Range Marks 1 MCQ GA A 1 2 MCQ GA C 1 3 MCQ GA B 1 4 MCQ GA B 1 5 MCQ GA B 1 6 MCQ GA A 2 7 MCQ GA D 2 8 MCQ GA D 2 9 MCQ GA B 2 10 MCQ GA C 2 1 MCQ XE A D 1 2 MCQ XE A A 1 3 NAT XE A 0.01 to 0.01 1 4 NAT XE A 5 to 5 1 5 NAT XE A 0.29 to 0.32 1 6 NAT XE A 250 to 250 1 7 NAT XE A 108 to 108 1 8 MCQ XE A B 2 9 MCQ XE A C 2 10 NAT XE A 3 to 3 2 11 NAT XE A 0.04 to 0.04 2 1 MCQ XE B B 1 2 MCQ XE B A 1 Q.No. Type Section Key/Range Marks 3 MCQ XE B D 1 4 MCQ XE B A 1 5 MCQ XE B C 1 6 MCQ XE B B 1 7 MCQ XE B A 1 8 NAT XE B 4 to 4 1 9 NAT XE B 39 to 42 1 10 MCQ XE B B 2 11 MCQ XE B C 2 12 MCQ XE B C 2 13 MCQ XE B A 2 14 NAT XE B 900 to 990 2 15 NAT XE B 40 to 42 2 16 NAT XE B 2.9 to 3.1 2 17 NAT XE B 107 to 127 2 18 NAT XE B 440000 to 442000 2 19 NAT XE B 0.001 to 0.002 2 20 NAT XE B 30 to 45 2 21 NAT XE B 4800 to 4800 2 22 NAT XE B 1200 to 1350 2 1 MCQ XE C C 1 2 MCQ XE C A 1 3 MCQ XE C B 1 Q.No. Type Section Key/Range Marks 4 MCQ XE C C 1 5 MCQ XE C A 1 6 MCQ XE C D 1 7 MCQ XE C C 1 8 NAT XE C 2 to 2 1 9 NAT XE C 6 to 6 1 10 MCQ XE C B 2 11 MCQ XE C A 2 12 MCQ XE C B 2 13 NAT XE C 430 to 500 2 14 NAT XE C 124 to 124 2 15 NAT XE C 2.60 to 2.70 2 16 NAT XE C 59.00 to 60.00 2 17 NAT XE C 1.85 to 2.00 2 18 NAT XE C 132 to 132 2 19 NAT XE C 5.01 to 5.17 2 20 NAT XE C 5 to 5 2 21 NAT XE C 15.5 to 18.5 2 22 NAT XE C 1420 to 1480 2 1 MCQ XE D D 1 2 MCQ XE D A 1 3 MCQ XE D D 1 4 MCQ XE D B 1 Q.No. Type Section Key/Range Marks 5 MCQ XE D B 1 6 NAT XE D 52.50 to 53.50 1 7 NAT XE D 27.00 to 28.00 1 8 NAT XE D 80 to 80 1 9 NAT XE D 220.00 to 222.00 1 10 MCQ XE D B 2 11 MCQ XE D B 2 12 MCQ XE D D 2 13 MCQ XE D C 2 14 MCQ XE D C 2 15 MCQ XE D A 2 16 MCQ XE D D 2 17 NAT XE D 195.00 to 197.00 2 18 NAT XE D 0.01 to 0.02 2 19 NAT XE D 14.5 to 15.5 2 20 NAT XE D 26.00 to 27.00 2 21 NAT XE D 3.10 to 3.20 2 22 NAT XE D 13.5 to 14.5 2 1 MCQ XE E B 1 2 MCQ XE E A 1 3 MCQ XE E C 1 4 MCQ XE E A 1 5 MCQ XE E C 1 Q.No. Type Section Key/Range Marks 6 MCQ XE E D 1 7 NAT XE E 160.0 to 161.0 1 8 NAT XE E 14.5 to 15.0 1 9 NAT XE E 0.30 to 0.32 1 10 MCQ XE E C 2 11 MCQ XE E C 2 12 MCQ XE E D 2 13 MCQ XE E B 2 14 MCQ XE E A 2 15 MCQ XE E D 2 16 NAT XE E 332.5 to 336.5 2 17 NAT XE E 3.01 to 3.07 2 18 NAT XE E 838.0 to 844.0 2 19 NAT XE E 961.0 to 962.0 2 20 NAT XE E 5.7 to 6.1 2 21 NAT XE E 12950.0 to 13000.0 2 22 NAT XE E 513.0 to 517.0 2 1 MCQ XE F C 1 2 MCQ XE F A 1 3 MCQ XE F D 1 4 MCQ XE F B 1 5 MCQ XE F D 1 6 MCQ XE F B 1 Q.No. Type Section Key/Range Marks 7 MCQ XE F A 1 8 NAT XE F 1500.00 to 1500.00 1 9 NAT XE F 48.52 to 48.72 1 10 MCQ XE F D 2 11 MCQ XE F A 2 12 MCQ XE F B 2 13 MCQ XE F D 2 14 MCQ XE F B 2 15 MCQ XE F D 2 16 MCQ XE F C 2 17 MCQ XE F C 2 18 NAT XE F 25.00 to 24.50 2 19 NAT XE F 98.80 to 99.10 2 20 NAT XE F 0.90 to 0.93 2 21 NAT XE F 17.91 to 18.00 2 22 NAT XE F 2600 to 2600 2 1 MCQ XE G B 1 2 MCQ XE G A 1 3 MCQ XE G C 1 4 MCQ XE G D 1 5 MCQ XE G D 1 6 NAT XE G 1.55 to 1.65 1 7 NAT XE G 103.0 to 103.2 1 Q.No. Type Section Key/Range Marks 8 NAT XE G 54 to 56 1 9 NAT XE G 0 to 0 1 10 MCQ XE G B 2 11 MCQ XE G C 2 12 MCQ XE G C 2 13 MCQ XE G A 2 14 MCQ XE G D 2 15 MCQ XE G A 2 16 MCQ XE G C 2 17 MCQ XE G B 2 18 NAT XE G 9.8 to 10.2 2 19 NAT XE G 1.1 to 1.8 2 20 NAT XE G 14.5 to 15.5 2 21 NAT XE G 6.0 to 6.4 2 22 NAT XE G 10 to 10 2 1 MCQ XE H B 1 2 MCQ XE H C 1 3 MCQ XE H A 1 4 MCQ XE H B 1 5 MCQ XE H B 1 6 MCQ XE H C 1 7 MCQ XE H D 1 8 MCQ XE H D 1 Q.No. Type Section Key/Range Marks 9 MCQ XE H B 1 10 MCQ XE H A 2 11 MCQ XE H B 2 12 MCQ XE H A 2 13 MCQ XE H B 2 14 NAT XE H 1254 to 1256 2 15 NAT XE H 19.42 to 19.44 2 16 NAT XE H 734.8 to 735.0 2 17 NAT XE H 9.16 to 9.18 2 18 NAT XE H 12.8 to 13.2 2 19 NAT XE H 92.4 to 92.8 2 20 NAT XE H 1.3 to 1.4 2 21 NAT XE H 0.11 to 0.12 2 22 NAT XE H 9.99 to 10.01 2

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