1. DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
UNIVERSITY OF NAIROBI
DEPARTMENT OF MECHANICAL AND MANUFACTURING ENGINEERING
ENGINEERING DESIGN II
FME 461
PART 1
GO NYANGASI
November 2009

2. DETERMINATION OF LOAD AND STRESS
LOAD CARRIED BY
MACHINE PART
STRUCTURAL PART

3. FREE BODY DIAGRAM
A machine or structural element can be removed from its position in the assembly or system,
Considered as a free-body, and
The influences of interacting parts , Slide 4
Replaced by forces and moments on the free-body, Slide 5

4. SPEED REDUCTION GEAR-BOX
Part 1 => Input Gear Shaft

5. INVOLUTE GEAR TOOTH PROFILE-FORCES

6. LOADING DIAGRAM: INPUT SHAFT-SPEED REDUCER

7. EQUILIBRIUM OF FREE BODY DETERMINATION OF LOADS
For an element in static equilibrium, or at constant velocity
Forces and moments that the interacting parts exert on the particular machine or structural part,
Represent the external influences of the interacting parts.
The part can then be analysed for static equilibrium, and
The equations of equilibrium written for forces and moments.
By solving the equilibrium equations,
The external loads carried by the machine or structural part are determined.

8. SPEED REDUCTION GEAR-BOX EXAMPLE
The example of a speed reduction gearbox is used to illustrate how the load carried by a gear shaft is determined.
This is done starting from the a priori decision based on requirements to be met by the object of design,
Stated as power (Kw) to be transmitted by the shaft, and
The speed (rpm) of power transmission.
A sketch of the speed reduction gearbox is shown in slide 4:

9. FORCES AND MOMENTS INPUT GEAR SHAFT
From the power and speed of transmission, the torque load (T) transmitted is determined.
From the torque load (T) transmitted and the diameter of the gears,
The tangential force (F) exerted at the pitch circle diameter is determined.
From the tangential force (F) on the gear wheel at the pitch circle diameter, and the pressure angle (20 degrees) of the gear teeth,
The resultant transverse force R=(Fsec 20) on the shaft is determined.
The load is simplified as a point load R=(Fsec 20) applied on a simply supported beam, and
The bending moment load (M) on the shaft is determined.

10. FORCES AND MOMENTS INPUT GEAR SHAFT

11. TORSION AND POINT LOAD INPUT GEAR SHAFT
By neglecting the effects of all other external loads,
The loading of the gear shaft can be simplified as
A solid circular shaft loaded in torsion, and simultaneously as
A straight beam (simply supported), loaded by transverse point load.

12. END DETERMINATION OF LOAD
Example: BENDING AND TORSION
Machine/Structural Part/Static equilibrium
Free body diagram
Forces and moments from interactions
Equations of equilibrium solved
Forces and moments determined
Case of torsion and bending of shaft

13. FORCES AND MOMENTS STRESSES INDUCED
SIMPLE TENSION
TORSION OF SOLID CIRCULAR SHAFT
SIMPLE BENDING WHEN DIRECT SHEAR IS IGNORED

14. STRESSES IN THREE DIMENSIONS
Three dimensional stress. Stresses on the third dimension are zero for plane stress

15. STRESSES IN TWO DIMENSIONS PLANE STRESS SITUATION
Two dimensional stress

16. BENDING AND TORSION SOLID CIRCULAR SHAFT
EXAMPLE OF PLANE STRESS
Refer to free body diagram Slide 5
Input shaft of the speed reduction gear
A case of a solid circular shaft subject to bending and torsion.
This stress situation occurs in many cases of loading of shafts

17. BENDING AND TORSION SOLID CIRCULAR SHAFT
In both bending and torsion, the extreme values of stress occur at the surface of the solid circular shaft.
Location of element subject to extreme values of stress ?
Surface of the shaft.
Let x direction of the element subject to plane stress coincides with the longitudinal axis of the shaft,
y direction is tangent to the surface of shaft,
z direction (where all stresses are zero), is normal to the surface of shaft.
This is illustrated in slide 14

18. BENDING AND TORSION SOLID CIRCULAR SHAFT
STRESS ELEMENTS

19. PLANE STRESS SITUATION
STRESS ANALYSIS- SLIDE 19-50
Starting with stress elements in a known plane
The outcome is determination of maximum and minimum stresses, and
The location of the planes on which extreme stresses occur

20. SIGNIFICANT STRESS MAX/MIN NORMAL AND SHEAR
Maximum and minimum normal stresses, and
Maximum and minimum shear stresses,
Are SIGNIFICANT STRESSES in the plane stress situation.
Magnitude of significant stresses are computed
By substituting the values of loads and dimensions
Stress elements on the member are determined.
Significant stresses are functions of stress elements

21. BENDING AND TORSION SOLID CIRCULAR SHAFT
Summary
Case of two dimensional (plane) stress
Stress elements
Shear stress due to torsion
Normal stress due to bending
Significant stress
maximum/minimum normal stress
Maximum/minimum shear stress

22. BENDING AND TORSION SOLID CIRCULAR SHAFT
STRESS ELEMENTS

23. TWO DIMENSIONAL STRESS GENERAL CASE
Stresses on the third dimension (z) are zero
The stress situation is two-dimensional stress or plane stress.
Situation is two normal stresses, one shear stress.
This situation occurs in many machine/structural parts.
This is illustrated in Figure 2.
DETERMINING SIGNIFICANT STRESS
Maximum/minimum normal stress
Maximum/minimum shear stress

24. FIGURE 2: PLANE STRESS SITUATION

25. STRESSES ON ARBITRARY PLANE
Starting with the known stresses on the x, y and z planes, it is necessary to determine the stresses on any other plane.
An arbitrary plane in the plane stress situation is shown in Figure 3,
Defined by
The angle by which the new plane has been rotated in the anti-clockwise (acw) direction,
Away from the x plane.

26. THE ARBITRARY PLANE
Stress situation on an arbitrary plane

27. FIGURE 3: STRESSES ARBITRARY PLANE

28. STRESSES ARBITRARY PLANE
The purpose of investigating the stress situation on any other plane
Is to determine how the stresses vary with the orientation of the plane away from the initial x, y, axis where stresses are known.
The unknown stresses on the variable plane are designated as a normal stress …. and a shear stress ….

29. PLANE STRESS TRANSFORMATION EQUATIONS
The stresses on the arbitrary plane
Is determined by
Resolving forces in the direction of the normal stress and
In the direction of the shear stress , and
Thereafter considering the equilibrium of the stressed element in the directions of normal stress , and shear stress .
This yields the plane stress transformation equations

30. PLANE STRESS TRANSFORMATION EQUATIONS
Normal and Shear stress on arbitrary plane

31. END Plane stress transformation equations
Stress on an arbitrary plane is determined
From these stress values on arbitrary plane
Magnitude of maximum stress and Location of plane of maximum stress, are determined
The proof is shown hereafter

32. TRANSFORMATION EQUATIONS TO SIGNIFICANT STRESSES
SLIDES 33 TO 49

33. PLANE STRESS TRANSFORMATION EQUATIONS
NORMAL STRESS
SHEAR STRESS

34. PLANE STRESS TRANSFORMATION EQUATIONS
The two equations transform the plane stress situation in the initial xy plane
To produce the stress situation in any other plane
Defined by the angle rotated anticlockwise from the initial x plane.
The unknown variable in the plane stress transformation equations
Is the location of the new plane defined by the angle of rotation

35. PLANE STRESS TRANSFORMATION EQUATIONS
The equations for normal and shear stresses on the arbitrary plane are therefore referred to as plane stress transformation equations.
They express the plane stress condition in any plane, in terms of the stress condition on the known plane, and
the location of the new plane relative to the known plane.
The result is that once the stress condition in one direction or plane are known,
The stress situation in any other direction
Defined by angle of rotation of the plane from the known plane
Can be determined from the plane stress transformation equations.

36. MAX/MIN NORMAL STRESS DETERMINATION
The maximum or minimum normal stress in the plane stress situation is determined by
Differentiating the plane stress transformation equation for normal stress with respect to the angle of rotation from the x plane,
Equating to zero, as shown in next slide

37. PRINCIPAL STRESS PLANE STRESS SITUATION
Differentiating Normal stress/plane location

38. Plane of Max/Min Normal Stress
Plane of maximum normal stress becomes
Solution

39. PRINCIPAL STRESS PLANE STRESS SITUATION
Values of the normal and shear stresses are already known,
The unknown location of the plane of maximum or minimum normal stress can be determined.
The maximum or minimum value of normal stress is determined
By substituting the angle given in previous slide
Into the plane stress transformation equation for normal stress .
This yields the values

40. MAXIMUM AND MINIMUM NORMAL STRESS

41. PRINCIPAL STRESS PLANE STRESS SITUATION
The maximum or minimum normal found by differentiating plane stress transformation equation in slide 33
Is referred to as the principal stress, and
The planes on which they occur as the principal planes.

42. MAXIMUM SHEAR STRESS IN DETERMINATION
Similarly, the location of maximum or minimum shear stress is determined

43. MAXIMUM SHEAR STRESS IN THE PLANE STRESS SITUATION
The location of plane of maximum or minimum shear stress becomes

44. MAXIMUM SHEAR STRESS IN THE PLANE STRESS SITUATION
The maximum or minimum value of shear stress in the plane is then determined
By substituting the value of angle location of plane obtained in slide 42 above
Into the plane stress transformation equation for shear stress .
This yields the values

45. MAXIMUM AND MINIMUM SHEAR STRESS
Maximum or minimum shear become

46. PRINCIPAL STRESSES
The principal stresses are the highest values of normal stress at a particular location of the loaded member
If failure is to occur as a result of normal stress, then
It will be caused by the principal stress and will occur on the principal plane.

47. MAXIMUM/MINIMUM SHEAR
Similarly, if failure is to occur as a result of shear stress,
Then it will be caused by the maximum or minimum shear stress
Will occur on the plane of maximum or minimum shear stress.

48. SIGNIFICANT STRESS IN THE PLANE STRESS SITUATION
The significant stresses in the plane stress situation are therefore
The principal (maximum and minimum normal) stresses, shown in next slide
The maximum and minimum shear stresses, shown in next slide

49. SIGNIFICANT STRESS PLANE STRESS
Summary RESULTS Significant stress

50. MOHR’S CIRCLE A GRAPHICAL METHOD DETERMINES SIGNIFICANT STRESSES
EXTREME VALUES OF
NORMAL STRESS
SHEAR STRESS
IN PLANE STRESS SITUATION
SLIDES 51-64

51. MOHR’S CIRCLE FOR PLANE STRESS
The plane stress transformation equations are parametric equations where the angle…. Of the plane is the parameter.
The two equations describe a circle drawn on the Cartesian coordinate system, with
Normal stress replacing x-axis,
Positive normal stress is drawn in the positive direction of the x-axis.
Shear stress replacing y-axis.
Convention is that clockwise shear is drawn in the positive direction of the y-axis.

52. STRESSES IN THE PLANE STRESS SITUATION

53. FIGURE 3: STRESSES ARBITRARY PLANE
Stress situation on an arbitrary plane

54. PLANE STRESS TRANSFORMATION EQUATIONS
Shear and Normal stress on any plane

55. DRAWING MOHR’S CIRCLE
Draw the xy-axis with normal stress in the x axis and shear stress in the y axis
Mark the value of stresses on x-plane (normal and shear) as a point on the x (normal stress) and y (shear stress) axes, observing the positive and negative directions of normal and shear stress.
Clockwise shear is positive and anti-clockwise shear negative.
This is point A.

56. DRAWING MOHR’S CIRCLE
Mark the value of stresses on y-plane (normal and shear) as a point on the, x and y axes, observing the positive and negative directions of normal and shear stress.
This is point B.
Join the points A and B

57. DRAWING MOHR’S CIRCLE
The point of intersection of the line AB, and the axis is the centre of Mohr’s circle C. The radius of Mohr’s circle is CB or CA.
Using the radius of the circle, draw Mohr’s circle for the plane stress situation.

58. DRAWING MOHR’S CIRCLE
The point of intersection of Mohr’s circle and the axis are the points of maximum and minimum normal stress.
The highest and lowest points of Mohr’s circle are the points of maximum and minimum shear stress, whose magnitude equals the radius of the circle.

59. DRAWING MOHR’S CIRCLE
Line CA is the direction of the x-axis in the stressed element, and the angle between CA and the axis is twice the angle separating the plane of maximum normal stress from the x-axis of the stressed element.
The location of the plane of maximum normal stress relative to the x direction in the stressed element, and that of all other planes, follows the positive shear rule.

60. DRAWING MOHR’S CIRCLE
The Mohr’s circle can be used as a fully graphical method for determining the stress in plane stress situation
By drawing the stress values to scale and
Measuring the values of the unknown stresses and their location relative to the xy-axes.

61. DRAWING MOHR’S CIRCLE
The method is better used as a semi-graphical method where the magnitude of stresses are found from the equations,
Only the angular location of the planes relative to the xy-axes are read from the , axes, as shown in the sketch below.

62. STRESSES IN THE PLANE STRESS SITUATION

63. FIGURE 3: STRESSES ARBITRARY PLANE
Stress situation on an arbitrary plane

64. DRAWING MOHR’S CIRCLE
Mohr’s circle for plane stress

65. END Summary Mohr’s circle for plane stress
Starting from plane stress transformation equations
Draw Mohr’s cicle

66. RELATIONSHIPS BETWEEN STRESS ELEMENTS
Principal planes are separated by an angle of 90o. Location of planes given by

67. RELATIONSHIPS BETWEEN STRESS ELEMENTS
Planes of maximum and minimum shear stress are separated by an angle of 90o
Similar to the case for normal stress
The maximum and minimum shear stresses are equal in magnitude but in opposite directions.

68. RELATIONSHIPS BETWEEN STRESS ELEMENTS
3) Shear stress on the principal plane= 0

69. Principal planes Planes of max/min shear stress

70. RELATIONSHIPS BETWEEN STRESS ELEMENTS
Normal stress on max/min shear planes

71. RELATIONSHIPS BETWEEN STRESS ELEMENTS
Value of normal stress on the planes
of maximum or minimum shear stress

72. Principal planes Planes of max/min shear stress

73. RELATIONSHIPS BETWEEN STRESS ELEMENTS
Principal planes and planes of maximum or minimum shear stress separated from by 45o

74. RELATIONSHIPS BETWEEN STRESS ELEMENTS
The maximum shear stress as a function of the principal stresses is given by

75. END
Summary
Relationships
Stress elements
Plane stress situation

76. THREE DIMENSIONAL STRESS GENERAL CASE
The three-dimensional stress situation is illustrated by the stresses on an element dx, dy, dz, in the Cartesian coordinate system.
The stress situation on the element is then as shown in the diagram in Figure 1.

77. FIGURE 1: ELEMENT IN THREE DIMENSIONS
Stressed Element

78. STRESSES AT A POINT
As the dimensions of the stressed element dx, dy, dz, are reduced to zero, the stresses on the element then represents the stresses at the point of the member where the stressed element is located.

79. EQUILIBRIUM OF STRESSED ELEMENT
The stressed element can be taken to represent a particular location of a loaded machine or structural part.
One of the axes of the stressed element shown in the Cartesian coordinate system can then be chosen to coincide with one of the dimensions of the part, eg the x axis to coincide with shaft axis.

80. EQUILIBRIUM OF STRESSED ELEMENT
The equilibrium of the three dimensional stressed element is fully described by three normal stresses, and six shear stresses.
Note that static equilibrium of the stressed element dictates that the shear stress in one plane is equal in magnitude but act in opposite direction to that in the next plane.

81. THREE DIMENSIONAL STRESS GENERAL CASE
PRINCIPAL STRESSES
The analysis of stresses carried out for plane stress situation can be extended to three dimensional stress situations where stresses occur on all the three axes of the stressed element.
The result of such analysis yields three principal stresses, and three principal planes.
The element is illustrated in the next slide

82. PRINCIPAL STRESSES THREE DIMENSIONAL STRESS

83. THREE DIMENSIONAL STRESS
The three principal stresses are then designated
To denote the order of magnitude and sign of the principal stresses, they are often designated as.

84. THREE DIMENSIONAL STRESS
MAXIMUM SHEAR STRESS
Extreme values of shear stresses in the three dimensional stress situation are then given in terms of the principal stresses as

85. THREE DIMENSIONAL STRESS
When the principal stresses have already been expressed in the order

86. THREE DIMENSIONAL STRESS
Then the maximum shear stress in the three-dimensional stress situation is given by

87. THREE DIMENSIONAL STRESS GENERAL CASE
Summary
Significant stresses
Three principal stresses
One maximum shear stress

88. UNI-AXIAL, BIAXIAL AND TRI-AXIAL STRESS SITUATION
When the stresses are expressed for the principal planes, then the principal stresses fully define the stress situation because the shear stresses are zero.
Stress situation expressed in terms of principal stresses are referred to as uni-axial, bi-axial, or tri-axial stress situation.

89. Uni-Axial Stress One dimensional (Normal/Shear)
Examples
Simple Tension or Compression
Simple Torsion
Pure Bending

90. Uni-Axial Stress One dimensional (Normal/Shear)

91. Bi-Axial Stress Two Dimensional (Plane)
Examples
Combined Bending and Torsion
Combined Tension and Torsion

92. Bi-Axial Stress Two Dimensional (Plane)

93. Tri-Axial Stress Three Dimensional stress
Examples
Thick Cylinder subject to internal pressure

94. Tri-Axial Stress Three Dimensional stress

95. UNI-AXIAL, BIAXIAL AND TRI-AXIAL STRESS SITUATION
The principal (maximum and minimum normal) stresses,
Maximum and minimum shear stresses,
Are significant stresses in many stress situations.
Maximum and Minimum shear stresses are determined from principal stresses

96. SIGNIFICANT STRESS VERSUS STRENGTH
Express stress condition as uni-axial, bi-axial or tri-axial
Express values of significant stress as principal stress or maximum shear stress,
Von-Mises stress expresses significant stress in terms of principal stresses.
Compare significant stress with the relevant indicator of strength to obtain factor of safety

97. THE END Design of machine/structural part-STRENGTH
Determination of load/loads on part
Location of extreme stress
Determination of Stresses
Determination of significant stress
Mohr’s Circle for plane stress
Determination of equivalent strength
Comparison of Stress and Strength
Determine size required for part or factor of safety for existing size