Lecture6-12-MEC223.pptxaegeeeeeeeeeeeeeeeeeeeeeeeeee

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1. MEC223 Design of Machine Element -I

2. Design Against Static Load 2 Stress Strain Relationship Shear Stress and Shear strain Stress due to bending moment Stress due to torsion moment Eccentric Axial Loading

3. Static Load A static Load is defined as a force, which is gradually applied to a mechanical component and which does not change its magnitude or direction with respect to time Three modes of failure Failure by elastic deflection Failure by general yielding; and Failure by fracture 3

4. Static Load Three modes of failure a) Failure by Elastic Deflection b) Failure by General Yielding c) Failure by fracture 4 The modu les of elasticity and rigidity are the important properties and the dimensions of the components are determined by load-deflection equations. The yield strength of a material is an important property. The Ultimate tensile strength of a material is an important property.

5. Nomenclature used.. A = Cross sectional area (mm 2 ) E = Young’s Modulus ( MPa or N/mm 2 ) G = Shear Modulus or modulus of rigidity ( MPa or N/mm 2 ) I = Moment of interia (mm 4 ) J = Polar Moment of inertia (mm 4 ) K = Bulk Modulus ( MPa or N/mm 2 ) M b = Applied Bending moment (N-mm) M t = Applied Torque (N-mm) 5

6. Nomenclature used.. P = External Force (N) Ssy = Yield Strength in shear ( MPa or N/mm2) Sut = Ultimate tensile strength ( MPa or N/mm2) Syt = Yield Strength of material ( MPa or N/mm2) U = Strain Energy (N-mm) ( fs ) = Factor of Safety K= Stiffness 𝛔 = Allowable Stress ( MPa or N/mm2) 𝛕 = Shear Stress ( MPa or N/mm2) 6

7. Factor of Safety (Safety Factor) In the calculations, Material strengths, Manufacturing process Is used to provide a design margin over the theoretical design capacity to allow for uncertainty in the design process. While Designing a component, It is necessary to provide sufficient reserve strength in case of an accident

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11. Factor of Safety (Safety Factor)

12. Factor of Safety (Safety Factor) The magnitude of factor of safety depends upon the following factors Effect of failure Type of load Degree of accuracy in force analysis Material of component Reliability of component Cost of component Testing of machine elements Service conditions Quality of manufacture

13. Factor of Safety (Safety Factor) FoS ( Based on yeild strength ) Application 1.25 – 1.5 Material properties known in detail Operating conditions known in detail Load and the resulting stresses and strains are known to a high degree of accuracy Low weight is important 2 – 3 For less tried materials or Brittle materials under average conditions of environment, load and stress 3 – 4 For untried materials under average conditions of environment, load and stress Better known materials under uncertain environment or uncertain stresses

14. Stress-Strain Relationship        

15. Assumptions for the Stress strain analysis The material is homogeneous The load is applied gradually The line of action of force P passes through the geometric axis of the cross section. The cross section is uniform. There is no stress concentration 15

16. Shear Stress and Shear Strain        

17. Shear Stress and Shear Strain Relationship between E,G and Poisson's ration   For carbon steels, m=0.29 For grey cast iron, m=0.21

18. Stresses Due to Bending Moment  

19. Assumptions for the Stresses due to bending moment The beam is straight with uniform cross section. The forces acting on the beam lie in a plane perpendicular to the axis of the beam. The material is homogeneous , isotropic and obey Hook’s law Plane cross sections remain plane after bending. 19

20. Sign conventions for Bending Moment

21. Stresses due to Torsional Moment    

22. Assumptions for Torsional moment The Shaft is straight with circular cross section. The plane transverse section remains plane after twisting The material is homogeneous , isotropic and obey Hook’s law 22 Calculation of torque from the power transmitted and the speed of rotation  

23. Eccentric Axial Loading

24. Principle Stresses (Mohr’s Circle) s x s y t xy Some Part A particular point on the part x y x & y  orientation

25. Mohr’s Circle t (CW) s x-axis y-axis s x s y t xy ( s x , t xy ) s x t xy s y t xy Center of Mohr’s Circle ( s y , t xy )

26. Requirement of Theories of Failures Theories of failure are used to determine the safe dimension of a component when it is subjected to combined stresses due to various loads. Theories of failure are used in design by establishing a relationship between stresses induced under combined loading conditions and properties obtained from tension test like S yt & S ut

27. Various Theory Of Failure Maximum principal stress theory ( Rankine’s Theory) Maximum shear stress theory (Guest and Treska’s Theory) Maximum Distortion energy theory (Von- mises and Henky’s Theory) Maximum principal strain theory (St. Venant’s theory) Maximum total strain energy theory (Haigh’ Theory)

28. Maximum Principle Stress Theory: Condition For Failure ): σ 1 S yt or S ut Condition for safe design: σ 1 < S yt /N or S ut /N Where N is factor of safety. For Tensile Stresses: For Compressive stresses:   The Theory states that the failure of the mechanical component subjected to bi-axial or tri-axial stresses occurs when maximum principal stress reaches the yield or ultimate strength of the material.

29. Region of safety

30. Maximum Principle Stress Theory This theory is suitable for the safe design of machine component made up of brittle material , because brittle material are weak in tension This theory is not good for design of ductile material because shear failure may occur

31. Maximum shear stress theory The Theory states that the failure of the mechanical component subjected to bi-axial or tri-axial stresses occurs when maximum shear stress at any point in the component becomes equal to the maximum shear stress in the standard specimen of the tension test, when yield starts.    

32. Maximum shear stress theory Condition For Failure: Condition for safe design:    

33. For tri-axial state of stress condition       The largest of the three stresses is equated to or      

34. Region of safety   For bi axial system ( =0)       Used for safe design for ductile material

35. Distortion Energy theory For biaxial stress The Theory states that the failure of the mechanical component subjected to bi-axial or tri-axial stresses occurs when strain energy of distortion per unit volume at any point in the component becomes equal to the strain energy of distortion per unit volume in the standard specimen of the tension test, when yield starts.    

36. 36 Region of safety

37. 37 Summary Theories of failures Maximum Principle Stress Theory Used for brittle materials with design criteria as   Maximum shear stress theory is used for safe design for ductile materials with design criteria as   Distortion energy theory is used for safe design for ductile materials with design criteria as (more accurate and complicated than Max. Shear stress theory)  

38. An Example Two Plates, Subjected to a tensile force of 50kN, are fixed together by means of three rivets as shown in fig (a). The plates and rivets are made of plain carbon steel 10C4 with a tensile yield strength of 250 N/mm 2 . Using maximum shear stress theory and factor of safety 2.5. Neglecting stress concentration, determine ( i ) the diameter of rivets; and (ii) the thickness of plates.

39. Cotter Joint A cotter joint is used to connect rigidly two co-axial rods or bars which are subjected to axial tensile or compressive forces . It is a temporary fastening .

40. Cotter Joint

41. Cotter Joint A cotter is a flat wedge shaped piece of rectangular cross section and its width is tapered (either on one side or on both sides(1 in 24)) from one end to another for an easy adjustment.

42. Notations Used P= tensile Load d= diameter of each rod d1= OD of socket d2= OD of spigot or ID of socket d3= diameter of spigot collar d4= diameter of socket collar a= distance from end of slot to the end of spigot b=mean width of cottar t= thickness of the cottar l= length of the cottar (all in mm)

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44. FBD of Forces

45. Possible Failure of Cotter Joint Component Tensile Failure Shear failure Crushing Failure (compressive) Bending Failure Rod Spigot Socket Cotter Component Tensile Failure Shear failure Crushing Failure (compressive) Bending Failure Rod YES NO NO NO Spigot YES YES YES NO Socket YES YES YES NO Cotter NO YES NO YES

46. Tensile Failure of Rods     = permissible tensile stress  

47. Tensile failure of Spigot   The thickness of the cottar is usually as t= .31 d  

48. Shear Failure of Spigot    

49. Crushing/Compressive Failure of Spigot    

50. Tensile Failure of Socket  

51. Shear Failure of Socket    

52. Crushing Failure of Socket    

53. Shear Failure of Cotter    

54. Bending Failure of Cotter          

55. Standard Proportions d1= 1.75d d2= 1.21d d3= 1.5d d4= 2.4d a=c= 0.75d b= 1.6d t= 0.31d t1= 0.45d Clearance= 1.5 to 3mm Taper for cotter= 1 in 32

56. Procedure for design Calculate diameter of each rod (d) Calculate thickness of cotter by empirical relation as t= 0.31d Calculate d2 of the spigot on the basis of tensile stress. Calculate OD d1 of the socket on the basis of tensile stress      

57. Procedure for design Calculate the diameter of spigot collar as(d3= 1.5d) and the socket collar (d4= 2.4d) Calculate a=c= 0.75d Calculate width b of cotter by considering shear and bending. Take the value larger from above considerations Check for the crushing and shear stresses in spigot and socket Calculate t1= 0.45d

58. It is required to design a cotter joint to connect two steel rods of equal diameter. Each rod is subjected to an axial tensile force of 50 kN. Design the joint and specify its main dimensions 58

59. Knuckle joint Two or more rods subjected to tensile and compressive forces are fastened together Their axes are not in alignments but meet in a point The joint allows a small angular moment of one rod relative to another It can be easily connected and disconnected, simple design and manufacuring Applications: tie bars in roof trusses, Chains, valve rods, Hand Pump etc

60. Knuckle Joint

61. Notations Used in Knuckle Joint D= Diameter of each rod (mm) D1 = Enlarged diameter of each rod (mm) d = diameter of knuckle pin (mm) do= outside diameter of eye or fork (mm) a= thickness of each eye of fork b= thickness of eye end rod-B (mm) d1= diameter of pin head (mm) X = distance of the center of fork radius R from the eye (mm)

62. FBD for Knuckle Joint and Tension Failure of Rod

63. Tensile Failure of Rods    

64. Shear Failure of Pin    

65. Crushing Failure of Pin in Eye  

66. Crushing Failure of Pin in Fork  

67. Bending Failure of Pin            

68. Tensile Failure of Eye  

69. Shear Failure of Eye    

70. Tensile and Shear failure of Fork Fork is a double eye Replace b with 2a to find corresponding equations    

71. Design Procedure for Knuckle Joint Calculate diameter of each rod Calculate enlarged diameter of rod end D1 = 1.1 D Calculate a and b as a = 0.75D and b = 1.25D Calculate diameters of pin by shear and bending and take maximum value from these      

72. Design Procedure for Knuckle Joint Calculate d = 2d and d 1 = 1.5d Check tensile, crushing and shear stresses for eye Check tensile, crushing and shear stresses for eye and fork The distance x is usually taken as 10mm            

73. Problem It is required to design a knuckle joint to connect two circular rods subjected to an axial tensile force of 80kN. The rods are co-axial and a small amount of angular movement between their axes is permissible. Design the joint and specify the dimensions of its components. Material used for two rods and pin is Grade 30C8 ( S yt = 400 N/mm 2 ). 73

74. Problem It is required to design a knuckle joint to connect two circular rods subjected to an axial tensile force of 80kN. The rods are co-axial and a small amount of angular movement between their axes is permissible. Design the joint and specify the dimensions of its components. Material used for two rods and pin is Grade 30C8( S yt = 400 N/mm 2 ). 74 Given P= 80 X 10 3 N Select factor of safety Calculation of permissible stresses Diameter of rods Enlarged diameter of the rod(D 1 ) Dimensions a and b Diameter of pin Dimensions d and d 1 Check for stress in eye and fork