Mechanics Of Solids Rr220304

Set No. 1

Code No: RR220304

II B.Tech II Semester Supplimentary Examinations, Apr/May 2007 MECHANICS OF SOLIDS ( Common to Mechanical Engineering, Production Engineering and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define and explain the terms: i. ii. iii. iv.

Modulus of Elasticity Modulus of Rigidity Poisson’s ratio Bulk Modulus.

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(b) Two vertical rods one of steel and the other of copper are each rigidly fixed at the top and 600 mm apart. The diameter and length of each rod are 30 mm and 375 mm respectively. A cross bar fixed to the rods at the lower ends carries a load of 5 kN such that the cross bar remains horizontal even after loading. Find the stress in each rod and the position of the load on the bar. ES = 200 Gpa and EC = 100 Gpa. [10] 2. (a) Derive the relationship between the three moduli of elasticity.

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(b) Show that in a prismatic bar, the maximum stress intensity due to a suddenly applied load is twice the stress intensity produced by the same load applied gradually. [8] 3. Sketch the shear force and bending moment diagrams showing the salient values for the loaded beam shown in the figure 3 below. [16]

Figure 3 4. (a) Discuss the assumptions involved in the theory of simple bending.

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(b) A cast iron beam has an I-section with top flange 100mm × 40mm, web 140mm×20mm and bottom flange 180mm × 40mm. If tensile stress is not to exceed 35MPa and compressive stress 95MPa, what is the maximum uniformly distributed load the beam can carry over a simply supported span of 6.5m if the larger flange is in tension. [10] 5. A beam ABC 13 m long is supported at A and B, such that AB = 10m and overhang BC = 3 m. It carries a point load of 4.5 kN from the end A and a 1 of 2

Set No. 1

Code No: RR220304

uniformly distributed load of 0.4 kN/m over the entire overhang. Determine: Slope at the end A, Deflection at the free end C and Maximum deflection ;Take E = 200 ×106 kN m2 and I = 3 ×10−5 m4 . [16] 6. A vertical steam boiler is of 2 m internal diameter and 5 m high. It is constructed with 20 mm thick plates for a working pressure of 1 N/mm2 . The end plates are flat and are not stayed. Calculate (a) the stress in the circumferential plates due to resisting the bursting effect and the stress in the circumferential plate due to the pressure on the end plates. [8] (b) the increase in length, diameter and volume. Assume the Poisson’s ratio as 0.3 and E = 200 GN/m2 . [8] 7. Derive an expression for the shear stress produced in a circular shaft which is subjected to torsion. What are the assumptions made in the above derivation ? [16] 8. A propeller shaft, 160mm external diameter, 80mm internal diameter, transmits 450kW at 4/3 Hz. There is, at the same time, a bending moment of 30kN-m and an end thrust of 250kN. Find (a) the maximum principal stresses and their planes

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(b) the maximum shear stress and its plane

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(c) the stress, which acting alone, will produce the same maximum strain. Take poisson’s ratio = 0.3 [4] ⋆⋆⋆⋆⋆

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Set No. 2

Code No: RR220304

II B.Tech II Semester Supplimentary Examinations, Apr/May 2007 MECHANICS OF SOLIDS ( Common to Mechanical Engineering, Production Engineering and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. An unknown weight falls 4 cm on to a collar rigidly attached to the lower end of a vertical bar 4m long and 8 cm2 in section. If the maximum instantaneous extension is found to be 0.42 cm, find the corresponding stress and the value of the unknown weight. E = 200 kN/mm2 . [16] 2. (a) What are the elastic constants ? Derive the relation between them.

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(b) A load of 2.0 kN is to be raised at the end of a steel wire. If the stress in the wire must not exceed 100 N/mm2 , what is the minimum diameter of the wire? What will be the extension in 5.0 m long wire? Take E = 210 kN/mm2 . [8] 3. (a) Define point of contra flexure.

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(b) Draw the SFD and BMD for the beam loaded as shown in the Figure3b. [13]

Figure 3b 4. (a) Derive from first principles the expression for shear stress at any point in a circular section of a beam when it is subjected to a shear force F. [8] (b) At a section of a rectangular beam the shear force is 165kN. The section of the beam is 120mm wide and 250mm deep. Calculate the shear stresses at 125mm, 50mm and 0mm from the neutral axis. Draw the sketch showing the variation of shear stress. [8] 5. (a) What is moment area method? Explain the two Mohr’s theorems, as applicable to the slope and deflection of a beam. [6] (b) A cantilever of uniform cross-section of length l carries two point loads, W at the free end and 2W at a distance a from the free end. Find the maximum deflection due to this loading. [10] 6. (a) Explain why ‘wire wound their cylinders’ are more efficient than ‘ordinary thin cylinders’. [6] 1 of 2

Set No. 2

Code No: RR220304

(b) A seamless pipe of 1m diameter is carrying a fluid under a pressure of 10 N/mm2 . Calculate the necessary thickness of the pipe, if the maximum allowable stress in the pipe material is 100 N/mm2 . [10] 7. (a) A rectangular body is subjected to direct stresses in two mutually perpendicular directions accompanied by a shear stress. Prove that the normal stress and shear stress on an oblique plane inclined at an angle θ with the plane of major direct stress, are given by [10] 1 1 σ n = /2 [σ x + σ y ] + /2 [σ x − σ y ]cos2θ + τ xy sin2θ τ s = 1/2 [σ x − σ y ] Sin2θ − τ xy cos2θ (b) A rectangular bar is subjected to a direct stress (σ) in one plane only. Prove that the normal and shear stresses on an oblique plane are given by σ n = σcos2 θ and τ s = σ/2 sinθ where θ = Angle made by oblique plane with the normal cross section of the bar. [6] 8. A circular shaft supported in bearings 4m apart transmits 75 kW power at 120 r.p.m. A pulley provided at 1.5 m from one bearing exerts a transverse load of 40 kN on the shaft. Determine a suitable diameter of the shaft if (a) The maximum direct stress is not to exceed 90 N/mm2 (b) The maximum intensity of shear stress is not to exceed 45 N/mm2 (c) The stress which acting alone, would produce the same maximum strain is not to exceed 90 N/mm2 (d) The direct stress which acting alone would produce the same maximum strain energy, is not to exceed 90 N/mm2 Take 1/m = 0.25. [4×4=16] ⋆⋆⋆⋆⋆

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Set No. 3

Code No: RR220304

II B.Tech II Semester Supplimentary Examinations, Apr/May 2007 MECHANICS OF SOLIDS ( Common to Mechanical Engineering, Production Engineering and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) The piston of a steam engine is 40 cm in diameter while the piston rod is 6 cm in diameter. The pressure of the steam acting is 1.05 N/mm2 . Find the stress in the piston rod and its elongation, if the piston rod is 75 cm long. E = 205 kN/mm2 when the piston is on in the instroke. [8] (b) A reinforced concrete column 50 cm in diameter has four 30 mm diameter steel rods embedded, and carries an axial load of 850 kN. Calculate the stresses in each of the two materials. E for steel = 2.04×105 N/mm2 and E for concrete = 0.136×105 N/mm2 . What is the adhesive force between steel and concrete. [8] 2. Prove that Poisson’s ratio for the material of a body is 0.5, if its volume does not change when stressed. Prove also that Poisson’s ratio is zero when there is no lateral deformation when a member is axially stressed. [16] 3. Sketch the shear force and bending moment diagrams showing the salient values for the loaded beam shown in the figure3 below. [16]

Figure 3 4. (a) A water main 110mm internal diameter is made of mild steel plate 12mm thick and is running full. If it is freely supported at the ends find the maximum permissible span if the bending stress is not to exceed 5MPa. Unit weight if steel is 81 kN/m3 and unit weight of water = 9.8 kN/m3 . [10] (b) State the assumptions involved in the theory of simple bending.

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5. (a) A beam AB of span l carries a distributed load of varying intensity from Zero at A to w per unit length at B. Measuring x from the end A, establish the equation for the deflection curve of the beam. [8] (b) A 3.5 meters long cantilever carries a uniformly distributed load over the entire length. If the slope at the free end is one degree, what is the deflection at the free end. [8]

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Set No. 3

Code No: RR220304

6. (a) Prove that the tendency to burst length wise is twice as great as a transverse section in a thin cylindrical shell subjected to an internal fluid pressure. [8] (b) A thin cylindrical shell 3 m long is of 1 m diameter. Determine the changes in length and diameter, if the shell is subjected to an internal pressure of 20 N/mm.2 . Take E = 200 kN/mm2 and 1/m = 0.28. [8] 7. (a) Define slenderness ratio. State the limitations of Euler’s formula. (b) Derive an expression for the Rankine’s crippling load for a column.

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(c) How will you justify the Rankine’s formula is applicable for all lengths of columns, ranging from short to long columns. [4] 8. A circular shaft supported in bearings 4m apart transmits 75 kW power at 120 r.p.m. A pulley provided at 1.5 m from one bearing exerts a transverse load of 40 kN on the shaft. Determine a suitable diameter of the shaft if (a) The maximum direct stress is not to exceed 90 N/mm2 (b) The maximum intensity of shear stress is not to exceed 45 N/mm2 (c) The stress which acting alone, would produce the same maximum strain is not to exceed 90 N/mm2 (d) The direct stress which acting alone would produce the same maximum strain energy, is not to exceed 90 N/mm2 Take 1/m = 0.25. [4×4=16] ⋆⋆⋆⋆⋆

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Set No. 4

Code No: RR220304

II B.Tech II Semester Supplimentary Examinations, Apr/May 2007 MECHANICS OF SOLIDS ( Common to Mechanical Engineering, Production Engineering and Automobile Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. A round steel bar 25 mm diameter and 360 mm long is placed concentrically within a brass tube which has an outside diameter of 35 mm and an inside diameter of 27.5 mm. The length of the tube exceeds, that of the bar by 0.15 mm. Rigid plates are placed on the ends of the tube, through which an axial compressive force of 80 KN is applied on the compound bar. Determine the compressive stresses in the bar and tube. E for steel = 2.1×105 N/mm2 . E for brass = 105 N/mm2 . [16] 2. Two parallel walls 6m apart are stayed together by a 25 mm diameter steel rod at 800 C passing through washers and nuts at ends. If the rod cools down to 220 C, calculate the pull induced in the rod, if (a) the walls do not yield and (b) the total yield at ends is 1.5 mm E steel = 2×105 N/mm2 , α steel = 11×10−6 per0 C. 3. (a) How do you classify loads? Give examples.

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(b) A simply supported beam of length 5m carries a uniformly increasing load of 800 N/m run at one end to 1600 N/m run at the other end. Draw the S.F. and B.M. diagrams for the beam. [12] 4. (a) A cantilever of length 3.2m fails when a load of 3kN is applied at the free end. If the cross section o f the beam is 50mm × 100mm. Find the stress at failure. [8] (b) A circular pipe of external diameter 80mm and thickness 8mm is used as a simply supported beam over a effective span 2.8m. Find the maximum concentrated load that can be applied at the centre of the span if the permissible stress in the tube is 140MPa. [8] 5. (a) A beam of length L is supported at each end with a couple applied at an intermediate point. Deduce an expression for the deflection and hence calculate the deflection at the point of application of the moment. [8] (b) A beam of length L carries a uniformly distributed load w/unit length and rests on three supports, two at the ends and one in the middle. Find how much the middle support be lower than the end ones in order that the pressures on the three supports shall be equal. [8] 1 of 2

Set No. 4

Code No: RR220304

6. (a) Prove that the tendency to burst length wise is twice as great as a transverse section in a thin cylindrical shell subjected to an internal fluid pressure. [8] (b) A thin cylindrical shell 3 m long is of 1 m diameter. Determine the changes in length and diameter, if the shell is subjected to an internal pressure of 20 N/mm.2 . Take E = 200 kN/mm2 and 1/m = 0.28. [8] 7. Derive an expression for the Euler’s crippling load for a long column with following end conditions: (a) both ends are hinged (b) both ends are fixed.

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8. A circular shaft supported in bearings 4m apart transmits 75 kW power at 120 r.p.m. A pulley provided at 1.5 m from one bearing exerts a transverse load of 40 kN on the shaft. Determine a suitable diameter of the shaft if (a) The maximum direct stress is not to exceed 90 N/mm2 (b) The maximum intensity of shear stress is not to exceed 45 N/mm2 (c) The stress which acting alone, would produce the same maximum strain is not to exceed 90 N/mm2 (d) The direct stress which acting alone would produce the same maximum strain energy, is not to exceed 90 N/mm2 Take 1/m = 0.25. [4×4=16] ⋆⋆⋆⋆⋆

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