BASIC OF MECHANICAL ENGINEERING (ME-1003)
PART-1 THERMAL ENGINEERING
BASIC CONCEPTS OF THERMODYNAMICS Theoretical Questions
1. Define the term thermodynamic. 2 Explain the thermodynamic system, surroundings and universe. 3. Distinguish between closed, open and isolated system with examples. 4. Differentiate between homogeneous and heterogeneous systems. S. What do you mean by phase of a system? 6. What do you mean by thermodynamic equilibrium?.. 9. Explain the terms: state, phase, process and cyclic process 10 What do you understand by reversible and irreversible process? Give examples. 11. Explain cyclic and quasi - static process. 12. Explain the term "Energy". Discuss its various forms. 13. Define work. Show that work is a path function. 14. Define the term heat. 15. Choose the open or closed system from the following: Water pump, pressure cooker, automobile engine, air compressor, steam turbine, boiler System. 16. State the number of phases in the following: - Alcohol mixture, coffee, water - oil mixture, aqua - ammonia, mild Pure water, water Steel, at room temperature. 17. Explain the concept of temperature. 18. State zeroth law of thermodynamics.
FIRST LAW OF THERMODYNAMICS Theoretical Questions 19. State first law of thermodynamics. 20. What is PMM-1? 21. Show that energy of an isolated system remains unchanged. 22. Write the steady flow energy equation. 23. Write the relationship between specific heats. 24. What is adiabatic index? Give its usual value. 25. What is a steady flow process?
26. What conditions are fulfilled by a steady flow process? '
Numerical Problems 27. 5 m3 of gas at 8 bars and 180°C is heated keeping the pressure same till the volume, Calculate (a) heat added, (b) external work done, and (c) change in intern 28. Energy during the process. [Ans. 14 MJ, 4 MJ, 10M 29. A jet planes is flying at a speed of 650 kmlh at an altitude where the pressure is 0.5 and temperature - l2°C. Air passes through the diffuser of the jet plane and leaves with velocity of 100mls. If the flow is isentropic, calculate the temperature and pressure at out let of the diffuser. [Ans. - 0.748°C, 0.5796bill 30. A gas flows through a turbine. At entrance its pressure is 7 bars and specific volumeO.m3/kg.At exit the corresponding values are 3.5 bars and 0.3 m3/kg.The velocity at inlet and exit is 150 mls and 300 mls respectively. Between entrance and exit the internal energy decreases by 80 kJlkg and 6 kJ/kg of heat is lost through radiation. Calculate the work developed per kg of gas flowing through the turbine. [Ans. 75.25 kJ/k1 31. Compressor requires 180 kJ/kg of work while the enthalpy of air increases by 73 kJIk as it passes through the compressor. The enthalpy of the circulating water increases by 5kJ/kg of air. Calculate the heat transfer from the air to the compressor.[Ans. 193 kJ/kg: 32. A gas turbine receives gases from the combustion chamber at 7 bar and 650°C, with velocity of 90 mls. The gases leave the turbine at 1 bar with a velocity of 45 mA Calculate the work done if the expansion is isentropic. Assume g = 1.333 anI Cp= 1.11 kJ/kg. [Ans. 397.45 kJ/kgJ 33. Duringasteadyflowprocess5000kg/hof fluid passes through an apparatus in which the xit pipe is 2m below the level of the inlet connection. The pressure decreases from 700to 140 kPa; velocity increases from 60 to 360 mls; internal energy decreases by 50 kJIkg; and specific volume increases from 0.06 to 0.36 m3/kg.The heat radiated from the apparatus is 10 MJI h. Calculate the work exchanged with the surroundings.[Ans. -32.475 kW] 34. A blower handles 1 kg/s of air at 20°C and consumes a power of 15 kW.The inlet and outlet velocities of air are 100mIsand 150mIsrespectively.Find the exit air temperature ,assuming adiabatic conditions. Take Cp=1.005 kJ/kg. for air; [Ans. 28.70°C] 35. A turbine operates under steady flow conditions, receiving steam at 1.2 MPa, 188°e, enthalpy 2785 kJlkg, 33.3 mIs and elevation 3 m. The steam leaves the turbine at20 kPa, enthalpy 2512 kJ/kg, 100 mls and elevation zero meter. Heat is lost to the surrounding the rate of 0.29 kJ/s. If thee rate of steam flow through the turbine is 0.42 kg/s, find the power output of the turbine in kW. [Ans. 112.515 kW] 36. At the inlet of a nozzle, the enthalpy of the passing fluid is 3000 kI/kg and velocity is60mls. At the discharge end, the enthalpy is 2762 kI/kg. The nozzle is
horizontal and the reis negligible heat loss from it. Find (a) velocity at exit, and (b) mass flow rate if inlet area is 0.1 m2 and specific volume at inlet is 0.187 m3/kg.[Ans. 692.5 mis, 32.08 kg/s] 37. A turbo compressor delivers 2.33 m% at 0.276 MPa, 43°C which is heated at this Pressure to 430°C and finally expanded in a turbine which delivers 1860 kW. During expansion, there is a heat transfer of 0.09 MIls to the surroundings. Calculate the turbine exhaust temperature assuming changes in kinetic and potential energy to be negligible.[Ans. 156.33°C] 38. A reciprocating air compressor takes in 2 m3/ininat 0.11 MPa, 20°C which it delivers at1.5MPa, 111°Cto an after cooler where the air is cooled at constant pressure to 25°C.Thepower absorbed by the compressor is 4.15 kW.Calculate the heat transfer in (a) the compressor, and (b) the cooler. [Ans. -0.1626 kJ/s, -3.768 kJ/s] 39. An adiabatic diffuser is employed to reduce the velocity of steam of air from 238 mls to50.4 mls. The inlet pressure is 1.02 bars and the outlet temperature is 394°C. Determine the final pressure and the required outlet area if the mass flow rate is 6.83 kg/so[Ans. 1.1715 bar, 0.23 m2] 40. Steam enters a heat exchanger at 2951.4 kJ/kg enthalpy where it condenses on the outside of heat-exchanger tubes. The condensed steam leaves the exchanger as a saturated liquid with 823.84 kJ/kg enthalpy and at a flow rate of 455 kglh. The steam is condensed bypassing cool water through the inside of the tubes. The cooling water experiences a temperature rise of 7.24°C before leaving the heat exchanger. What flow rate of cooling water is required in kglh ? [Ans. 31988 kglh] 41. A gas flows steadily through a rotary compressor. The gas enters the compressor at a temperature at 16°C, a pressure 0('100 kPa, and an enthalpy of 391.2 kJ/kg. The gas leaves the compressor at a temperature of 245°C, a pressure of 0.6 MPa, and an enthalpy of 534.5 kJ/kg. There is no heat transfer to or from the gas as it flows through the compressor. Evaluate (a) the external work done per unit mass of gas assuming the gas velocities at entry and exit to be negligible, and (b) the external work done per unit mass of gas when the gas velocity at entry is 80 mls and that at exit-is 160 mls., [Ans. 143.3 kJ/kg, 152.9 kJ/kg] 42. The steam supply to an engine comprises two steams which mix' before entering the engine. One stream is supplied at the rate of 0.01 kg/s within enthalpy of 2952 kJ/kg and velocity of 20 mls. The other stream is supplied at the rate of 0.1 kg/s with an enthalpy of 2569 kJ/kg and a velocity of 120mls. At the exit from the engine the fluid leaves as two streams, one of water at the rate of 0.001 kg/s with an enthalpy of 420 kJI kg and the other of steam; the fluid velocities at the exit are negligible. The engine develops shaft power of 25 kW.The heat transfer is negligible. Evaluate the enthalpy of the seconde"it stream. [Ans. 2401 kJ/kg]
APPLICATION OF FIRST LAW OF THERMODYNAMICS
Theoretical Questions 45. Determine an expansion for the heat transfer in a closed system isochoric process. 46. Explain the concept of a closed isobaric process and determine an expansion for its heat Transfer. 47. Determine an expansion for the work done in a closed isothermal process. 48. What is an isentropic process? Determine an expansion for the work done in a non-flow 49. Isentropic process. 50. Determine an expansion for the heat transfer and work done in a non-flow Polytrophic 51. Process. I 52. What is a free expansion process? What are its characteristics? 53. What is a throttling process? State its characteristics? 54. What is a steady 55. Flow process? Determine the work done in such an isochoric process. 56. Determine an expansion for the work done in a steady flow adiabatic process. 57. Compare the work done in a non-flow and flow type polytrophic process.
Numerical Problems 45. 01 kg of air at 3.5 bar and occupying 0.35 m3 is heated at constant volume until its Temperature has rises to 316°C. Find (a) heat added, and (b) gain in internal energy per kg of air. Take Cv =0.718 kJ/kg. K. [Ans. 116.44kJ/kg, 116.44 kJ/kg ] 46. 1kg of air having an initial volume of 0.3 m3 is heated at constant pressure of 3.2 bar until the volume is doubled. Find (a) heat added, and (b) work done. Take Cp =1.005kJ/kg.K, Cv =0.718 kJ/kg.K. [Ans. 336.17 kJ, 96 kJ] 3.3 5 m3of gas at 8 bar and 180°C is heated keeping the pressure same throughout till the volume is doubled. Find the heat added, external work done and the change in internal energy during the process. Take Cp=1.005 kJ/kg. K, CV= 0.718kJlkg. K.[Ans. 14006.97 kJ, 4000kJ, 10008.06 kJ] 47 Calculate the difference in non-flow workup and flow work -f v.dp, if lkg of air having specific volume of 0.4 m3 at 1.35 bar is compressed to a pressure of 10 bar according to the law pV 1.3= C. [Ans. 31.7 kJ] 48 gas undergoes the following reversible non-flow processes during its initial pressureof15bar and volume 1 m3/kg to final pressure of 6 bar and volume 2 m3/kg. (a) Constant volume followed by constant pressure. (b) Constant pressure followed by constant volume. (c) Polytrophic process, pV1.3= c. Calculate the work done by the gas during each process.
47. [Ans. 600 kJ/kg, .1500 kJ/kg, 1000 kJ/kg] (a) A certain gas occupies 0.1 m3 at 1bar pressure. It is compressed adiabatically to a pressure of 7bar.Find the new volume, and change in internal energy. The density of gas at O°Cand I bar is 1.5 kglm3. Take Cv= 0.728 kJ/kg. K. [Ans. 0.0249 m3, 18.847 kJ]3.7 3kg of oxygen undergoes anon-flow process during which its pressure remains constant at 7 bar from an initial volume of 1200 liters to a state where the temperature is 520 K.Find (a) changes in internal energy, (b) change in enthalpy, (c) heat transferred, and (d)work done. For oxygen, Cv =0.661 kJlkg. K, Cp =0.921 kJ/kg. K.[Ans. -1104.38 kJ, 1538.78 kJ, - 1538.78 kJ, - 434.4 kJ] (b) A constant volume chamber of 0.3 m3 capacity contains 2 kg of a gas at 5°C. Heat is transferred to the gas until the temperature is 100°C. Find the work done, heat transfered, and the change in internal energy, enthalpy and the entropy. Take Cp=1.968kJ/kg.K and1.507 kJ/kg.K [Ans. 0, 286.33 kJ, 286.33 kJ, 373.92 kJ, 0.886 kJlK]Cv=60°C to 0.4 MPa, 48. 0.5kg of air is compressed reversibly and adiabatically from 80 kPa, 49. And is then expanded at constant pressure to the original volume. Calculate the heat R = 0.287 kJ/kg. K, Cp= 1.005 kJItransfer and work transfer for the whole path. Take kg. K and CV = 0.718 kJ/kg. K. [Ans. 571.64 kJ, 93.5 kJ]
SECOND LAW OF THERMODYNAMICS Theoretical Questions 55. State the limitations of First law of thermodynamics. 56. State Kelvin-Planck statement for second law of thermodynamics. 57. State Clausius statement for the second law of thermodynamic. 58. What is a PMM2? 59. Are Kelvin-Planck and Clausius statements equivalent? 60. Differentiate between a heat engine and a heat sink. 61. What is a thermal reservoir? 62. Differentiate between a heat source and a heat sink. 63. Describe the working of a Carnot cycle. 64. What are the limitations of Carnot cycle? 65. Define thermal efficiency of a heat engine. 66. Differentiate between a heat pump and a refrigerator. 67. Define COP of a heat pump and a refrigerator. 68. State Carnot theorem. 69. What is Clausius inequality? 70. State Carnot's theorem. 71. Define the term 'Entropy'.
72. Show that the entropy is a property of the system. 73. The entropy of the universe tends to be maximum. Comment. 74. What are the characteristics of entropy? 75. Draw the Carnot cycle on T-S diagram. 76. State Third law of thermodynamics. 77. What is the importance of third law of thermodynamics?
Numerical Problems 78 A heat engine receives 1000kW of heat at constant temperature of 285°C and rejects heat at 5°C. The possible heats rejected are: (a) 840 kW, (b) 442 kW and (c) 300 kW.Comment on the results. [ADS. Not a possible cycle] 79 To cool water for drinking during summer, 1 kg of ice at - 2°C is mixed with 5 kg of water at 27°C in an insulated container subjected to 1bar atmospheric pressure. Calculate~ change in temperature of water and the change in entropy of mixture. Take specific heat of ice as 2.093 kJ/kg K and the latent heat of ice as 335 kJ/kg. Specific heat of water is 4.187 kJ/kg K. [ADS.9.817°C, 0.0907 kJ/K] 80 A heat exchanger handles 40 kg/ min of water which is heated from 20°C to 70°C by a hot gas entering the heat exchanger at 150°C and flow rate 80 kg/min.Determine the change in entropy. Assume specific heat for water as 4.18 kJ/kg K and for gasses 1.045kJ/kgK. , [ADS.0.0632 kW/K] 81 A heat pump operates between two identical bodies of specific heat Cat 'The operation of the pump cools down one of the bodies to Tz.Show that for ‘the operation of pump the minimum work input is given by: T.zW. = C 1nun Tz + T2 -'- 27)][ 82 1 Kg. of air is compressed according to the law pV1.25=constant from a pressure of 1.03bar and temperature 15°C to a pressure of 17 bar. Calculate (a) temperature at the end of compression,(b) heat rejected or received during the process, and(c) change in entropy. Take Cp = 1.005 kJ/kg K and Cv =0.715 kJ/kg K [ADS.504.56 K, - 96.368 kJ/kg, - 0.2495 kJ/kg K] 4.6 4 kg of air is compressed in a reversible steady flow polytrophic process, pVl.25=C, from 1bar and30°C to 10bar.Calculate the work input, heat transferred and change in entropy. [ADS.1017.24 kJ, - 304.81 kJ, - 0.793 kJ/K] 83 Anadiabaticvesselcontains2kgofwaterat25°C.Bypaddle-wheelworktransfer, the Temperature of water is increased to30°e. If the specific heat of wateris4.187kJlkgK, find the entropy change of the universe. [ADs.0.139 kJlK] 84 Two blocks of metal, each of mass 10 kg and a specific heat of 0.4 kJ/kg K, are at at temperature of 40°C. A reversible refrigerator receives heat from one block and rejects heat to the other. Calculate the work required to cause a temperature difference of 100°Cbetweenthetwoblocks. [ADs.32kJ]
PATR -2
STRENGTH OF MATERIAL
SIMPLE STRESS AND STRAIN Theoretical Questions 1 2. 3. 4. 5.
6.
7.
8. 9, 10.
Define stress and strain. Write down the S.I. and M.K.S. units of stress and strain. Explain clearly the different types of stresses and strains. Define the terms: Elasticity, elasticlimit, Young'smodulusand Modulus of rigidity. State Hooke's law. Three sections of a bar are having different lengths and different Diameters. The bar is subjected to an axial load p. Determine the total change in length of the bar. Take Young's'modul of different sections same. Distinguish between the following, giving due explanation: (1) Stress and strain, (2) Force and stress, and (3) Tensile stress and compressive stress. Prove that the total extension of a uniformly tapering rod of Diameters D1 and D2, whenthe rod issubjectedto an axial load P is given by 4PL dL . Ceded Where L=Total length of the rod Define a composite bar. How will you find the stresses and load Carried by each member of a composite bar? Define modular ratio, thermal stresses, thermal strains and Poisson’s ratio. A rod whose ends are fixed to rigid supports, is heated so that Rise in temperature is rc. Prove that the thermal strain and stresses Setup in the rod are given by, Thermal strain=a. T and Thermal stress=a.T.E. Where a=Co-efficient of linear expansion. 11. What is the procedure of finding thermal stresses in a composite?
12 13
14
Bar? what do you mean by 'a bar of uniform strength’? Find an expression for the total elongation of a bar due to its Own weight, when the bar is fixed at its upper end and hanging freely at the lower end. Find an expression for the total elongation of a uniformly Tapering rectangular bar when it is subjected to an axial load P.
Numerical Problems 1. A rod 200 cm long and of diameter 3.0 cm is subjected to an axial pull of 30 kN.If the Young’s modulus of the material of the rod is2 N/mm, determine: (i) stress, (2) strain and (3) the elongation of the rod. [Ans. (i) 42.44 N/mm2 (2) 0.000212 (3) 0.0424 cm] 2. Find the Young’s modulus of a rod of diameter 30mm and of Length 300mm which is subjected to a tensile loadof60kNandthe extension of the rod is equal to 0.4 mm. [Ans. 63.6 GN/m2] 3. The safe stress, for a hollow steel column which carries an axial load of 2.2x103 kN is 120MN/m2.If the external diameter of the column is 25 cm, determine the internal diameter. [Ans. 19.79 cm] 4. An axial pull of 40000N is acting on a bar consisting of three Sections of length 30 cm, 25 cm, and 20 cm and of diameters 2 cm, 4 cm and 5 cm respectively. If the Young's modulus=2xtOSN/mm2, determine. l) Stress in each section and (u) total extension of the bar. [Ans. (i) 127.32, 31.8, 20.37N/mm2, (ii) 0.025 elf 5. The ultimate stress for a hollow steel column which carries an Axial load of 2 MN is 500N/mm2.If the external diameter of the column is 250mm, determine the internal diameter. Take the factor of safety as 4.0. [Ans. -205.25 mm]
SHEAR FORCES AND BENDING MOMENT DIAGRAM Theoretical Questions 6. Define and explain the following terms: Shear force, bending moment, shear force diagram and bending Moment diagram. 7. What are the different types of beams? Differentiate between a cantilever and a simply supported beam. 8. What are the different types of loads acting on a beam? 9. Differentiate between a point load and a uniformly distributed load.
10. What are the. Sign conventions for shear force and bending Moment in general? ' 11. Draw the S.F. and, B.M. diagrams for a cantilever of length L Carrying a point load W at the free end. 12. Draw the S.F. and B.M. diagrams for a cantilever of length L Carrying uniformly distributed load of w per m length over its entire length. 13. Draw the S.F. and, B.M. diagrams for a cantilever of length L Carrying a gradually varying load from zero at the free end to w per unit Length at the fixed end. 14. Draw the S.F. and B.M. diagrams for a simply supported beam of length L carrying a point loads Wat its middle point. 15. Draw the S.F. and B.M. diagrams for a simply supported beam Carrying a uniformly distributed load of w per unit length over the entire Span. Also calculate the maximum B.M. 16. Draw the S.F. and B.M. diagrams for a simply supported beam Carrying a uniformly varying load from zero at each end to w per unit length at the centre. ' 17. What do you mean by point of contra flexure? Is the point of Contra flexure and point of inflexion different? 18. n. How many points of contra flexure you will have for simply Supported beam overhanging at one end only? 19. How will you draw the S.F. and B.M. diagrams for a beam? Which is subjected to inclined loads? What do you mean by thrust diagram? 20. Draw the S.F. and B.M.diagramsfor a simply supported beam of length L which is subjected to a clockwise couple 11at the centre of the Beam.
Numerical Problems 21. A cantilever beam of length2 m carries a point load of 1 KN At its free end, and another load, of 2 kN at a distance of 1m from the free End. Draw the S.F. and B.M. diagrams for the cantilever. [Ans. Fnuu=+3kN; Mmax=-4kNm] 22. A cantilever beam of length 4 m rise' point loads of 1 kN, 2 KN and 3 KN at 1, 2 and 4 m from the fixed end. Draw the shear force an~ R.M. diagrams for the cantilever. [Ans. F...=+6 KN; M...=-17 kNm 23. A cantilever of length2 m carries a uniformly distributes load of 3 kN/m run over a length of 1m from the fixed end. Draw the S.F and B.M. diagrams.' [Ans. FmlU'=+3kN; Mmax=-1.5 kNm] 24. A cantilever of le~gth5, m carries a uniformly distributed load of 2 kN/m length over the whole length and a point load of 4 kN at the free end. Draw the S.F. and B.M. diagrams for the cantilever. [Ans. Fmcx=+14kN; Mmax=-45kNm] 25. A cantilever 2 m long is loaded with uniformly distributed load of 2 kN/m run over a length of 1m from the free end. It also Carpes
a point load of 4 kN at a distance of 0.5 m from the free end.-Draw the Shear force and B.M. diagrams. [Ans. FmlU'=+6kN; Mmax""'; 9kNm) 26. J A cantilever of lenE,'th6 m carries two point loads of 2 KN ~hd 3 KN at a distance of 1 m and 6 m from the fixed end respectively in addition to this the beam also carries a uniformly distributed load of 1kN/m over a length of 2 m at a distance of 3 m from the fixed end. Draw the S.F. and B.M. diagrams. [Ans. FmlU'=+7kN; Mmax=-2{!kNm] 27. A cantilever of length6mcarries a gradually varying load, zero at the free end to 2 kN/mat the fixed end. Draw the S.F. and B.M diagrams for the cantilever. [Ans. -Finmc=+6kN; Mmax=-12kNm] 28. A simply supported beam of length 8 m carries point loads of 4 kN and 6 kN at a distance of 2 m and 4 m from the left end. Draw the S.F:.and B.M. diagrams for the beam. [Ans. Mmax::,+20kNm] 29. A simply supported beam of length 10m carries point loads of 30 kN and 50 kN at a distance of 3 m and 7 m from the left end. Draw the S.F. and B.M. diagrams for the beam. [Ans. Mmax=+132kNm] 30. A simply supported beam of length 8 m carries point loads of 4 kN, 10kN and 7 kN at a distance of 1.5 m, 2.5 m and 2 m respectively From left end A. Draw the S.F. and B.M. diagrams for the simply supported beam. [Ans. Mmax=+90kNm] 31. A simply supported beam is carrying a uniformly distributed load of 2 kN/m over a length of 3 m from the right end. The length of the beam is 6 m. Draw the S.F. and B.M. diagrams for the beam and also calculate the maximum B.M. on the section. [Ans. Mmax=+5.06kNm] 32. A beam of length 6 m is simply supported at the ends and carries a uniformly distributed load of 1.5kN/mrun and three co ncentrated loads of 1 kN, 2 kN and 3 kN acting at a distance of 1.5 m, 3 m and 4.5 m respectively from left end. Draw the S.F. and B.M. diagrams and Determine the maximum bending moment. . [Ans. 12.75kNm]
BENDING STRESS IN BEAM Theoretical Questions 33. Define the terms: bending stress in a beam. neutral axis and section modulus. 34. What do you mean by 'simple bending' or 'pure bending’? 35. What are the assumptions made in the theory of simple bending? 36. Derive an expression for bending stress at a layer in a beam. 37. What do you understand by neutral axis and ", moment of Resistance? 38. Prove the relation, M=L=~
I y R' l=M.O.I. where M=Bending moment, y=Distance from N.A. f=Bending stress, R=Radius of curvature. E=Young's modulus, and (Bangalore University, Jan. 1990) 39. What do you mean by section modulus? Find an expression for section modulus for a rectangular, circular and hollow circular sections. 40. How would you find the bending stress in unsymmetrical section? 41. What is the meaning of 'Strength of a section’? 42. Define and explain the terms: modular ratio, fletched beams and equivalent section. 43. What is the procedure of finding bending stresses in case of fletched beams when it is of (i) a symmetrical section and (ii) an unsymmetrical section? 44. Explain the terms: Neutral axis, section modulus, and moment of resistance. (Bangalore University, July 1988) 45. Show that for a beam subjected to pure bending, neutral axis coincides with the centroid of the cross-section. (Bangalore University, March 1989) 46. Prove that the bending stress in any fiber is proportional to the distance of that fiber from neutrally in beam. (BhavnagarUniversity, 1992)
Numerical Problems 47. A steel plate of width 60 mm and of thickness 10 mm is bent into a circular arc of radius 10 m. Determine the maximum stress induced and the bending moment which will produce the maximum stress. Take E=2xIOS N/mm2. [Ans. 100 N/mm2 ; tOONm] 48. A cast iron pipe of external diameter 60 mm, internal diameter of 40 mm, and of length 5 m is supported at its ends. Calculate the maximum bending stress induced in the pipe if it carries a point load of 100 N at is centre. [Ans. 7.34 N/mm] 49. A rectangular beam 300 mm deep is simply supported over a span of 4 m. What uniformly distributed load per meter, the beam may carry if the bending stresses isn.ott.oexceed 120N/mm2? Take 1=8xl06 mm4. [Ans. 3.2kN/m] 50. A cast iron cantilever .of length 1.5 meter fails when a point load W is-applied at the free end. If the 5ectiJin.ofthe beam is 40 mmx60 mm and the stress at the failure is 120N/mm\ find the point load applied. [Ans. 1.92 kN] 51. A cast iron beam 20 mmx20 mm in section and 100cm long is-
Simply supported at the ends. It carries a point load w at the centre. The Maximum stress induced is 120N/mm2.What uniformly distributed load will break a cantilever .of. the same material 50 mm \vide, 100 mm deep and 2 m long? [ADs. 5 kN per m run] 52. Atimberbeamis120romwideand200romdeepandisused.onaspan of 4 metreS.The beam carries a uniformly distributed lood.of 2.8kNlmrun.overthe Entire length. Find the maximum bending stress induced. [ADs. 7 N/mm2] 53. A timber cantilever200mmwide and 300mm deep is3m long. It is loaded with a U.D.L.of3 kN/m .over the entire length. A point load Of2.7 kN is placed at the free end .of the cantilever. Find the maximum bending stress produced. [Ans. 7.2 N/mm2] 54. A timber beam is freely supp.orted.onsupp.orts6 m apart. It carries a uniformly distributed l.oad.of12kN/m run and a p.ointl.oad.of9 kN at 3.5 m from the right supP9rt. Design suitable section. Of the beam making depth twice the width, if the stress in timber is not to exceed 8 N/mm2. [Ans. 230 mmx460 mm]
TORSION Theoretical Questions
55. Define the terms: Torsion, torsion rigidity and polar moment of inertia. 56. Derive an expression for the shear stress produced in a circular shaft which is subject to torsion. What are the assumptions made in the derivation? 57. Prove that the torque transmitted by a solid shaft when Subjected to torsion is given by T=!! J, D3 16 where D=Dia. of solid shaft and /. =Max. Shear stress. 58. When a circular shaft is subjected to torsion show that the shear Stress varies linearly from the axis to the surface. (AMIE Winter, 1982) S. Derive the relation for a circular shaft when subjected to torsion as given below !..=h.=C9 JRL where T=Torque transmitted, J=Polar moment of inertia, /.= Max. Shear stress, R=Radius of the shaft, C= Modulus rigidity,
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()=Angle of twist, and L= I, length of the shaft. 59. Find an expression for the torque transmitted by a hollow Circular shaft of external diameter=Do and internal diameter=Dj. 60. Define the term 'Polar modulus'. Find the expressions for polar Modulus for a solid shaft and for a hollow shaft 61. What do you mean by 'strength of a shaft' 1 62. Define torsional rigidity of a shaft. Prove that the torsional rigidity is the torque required to produce a twist of one radian in a unit length of the shaft. 63. Prove that the strain energy stored in a body due to shear stress is given by, 2 U=.!Lxv 2C where q=Shear stress, C= Modulus of rigidity, and V=Volume of the body. 64. Find an expression for strain energy stored in a body which due to torsion or Prove that the strain energy stored in a body due to torsion is given By, 2 U=.Lxv 4C where q=Shear stress on the surface of the shaft, C=Modulus of rigidity, and V=Volume of the body. 65. A hollow shaft of external diameter D and internal diameter d is subjected to torsion, prove that the strain energy stored is given by, 2 U=~ (D2+cf) xV 4CD where V=Volume of the hollow shaft and q=Shear stress on the surface of the shaft. 66. What is a spring? Name the two important types of spring.
Numerical Problems 67. A solid shaft of 20 cm diameter is used to transmit torque. Find the maximum torque transmitted by the shaft if the maximum shear stress in duced in the shaft is 50N/mm2. . [Ans. 78539.8 Nm] 68. The shearing stress in a solid shaft is not to exceed 45 N/mm2 When the torque transmittedis40000N-m. Determine the minimum diameter of the shaft. [Ans. 16.49mm] 69. Find the maximum torque transmitted by a hollow circular shaft
of external diameter 30cm and internal diameter 15em if the shear stress Is not to exceed 40 N/mm2. [Ans. 198.8 kN] 70. Two shafts of the same material and of same lengths are Subjected to the same torque, if the first shaft is of a solid circular section and the second shaft is of hollow circular section, whose internal diameter !s0.7 times the outside diameter and the maximum shear stress developed In each shaft is the same, the weights of the shafts. 1.633
71. Find the maximum shear stress induced in a solid circular shaft Of diameter 20 cm when the shaft transmit 187.5kWat 200r.p.m. [Ans. 5.7 N/mm2] 72. A solid circular shaft is to transmit 375kWat 150r.p.m. (I) find the diameter of the shaft if the shear stress is not to exceed 65 N .
