Centroid and Moment of Inertia from mechanics of material by hibbler related to the subject of materials in mechanical engineering

Mechsnics Of Material by Hibbler

1. Centroids & Moment of Inertia MMT -I 1

2. Centroid or Center of Area Centroid : A point at which whole area of body or a plane figure is acting is called centroid. Also known as geometric center It is only applicable to plane figures having certain area but no volume, for example; rectangle, square, circle, semi-circle, triangle, etc. Composite Figure made up of two or more plane figure   such as triangles, rectangles, circles, semi-circles, etc Center of Gravity (COG) point at which the whole mass of body is concentrated . COG and centroid are different quantities but both become similar when bodies have only area but not weight. 2

3. Plane Figures vs Composite Figures 3

4. Centroid vs Center of Gravity (COG) Whole weight is acting at this point Other Examples = Book or pen balanced on finger Suspend the plane fig from corners From suspension point, draw a vertical line that will be directed towards the earth Repeat the step at other corners also. Intersection point of lines will be the centroid of fig 4

5. Centroid vs Center of Gravity (COG) Steel Plate Steel +Aluminum Plate COG Centroid Centroid COG Entire area of plane figure is concentrated Entire weight is assumed to be concentrated 5

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7. Centroid Location of a composite figure Place x axis at lowest point and y axis at the left edge of figure (first quadrant, where X & Y will be + ive ). Split the composite figure into plane figure & find centroid of each plane figure Now following formulas can be used to find the X and Y coordinates of centroid of composite figure . ,   x & y in above formula mean the dimensions along x & y axis respectively. Locate the centroid on figure 7  

8. Figure having one axis of symmetry : Any figure having one axis of symmetry , its centroid will lie at that axis . Centroid of figure will lie on x-axis because it is symmetric about x axis. Therefore only one coordinate must be calculated to locate the centroid. Figure having two axis of symmetry : Any figure having two axis of symmetry , then its centroid will lie at the intersection of axis of symmetry . 8

9. Figure having no axis of symmetry : Any figure having no axis of symmetry , its centroid will be located by inspection . Such figure is only symmetric about a point, called the center of symmetry, such that every line drawn through that point contacts the area in a symmetrical manner. 9

10. Q. Determine the centroid of C section beam. X 3 X 2 X 1 y 3 y 2 y 1 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 7.5 2.5 0.75 10.5 0.75 5 7.5 2.5 9.25 Area X-coordinate Value (cm) Y-coordinate Value (cm) 7.5 2.5 0.75 10.5 0.75 5 7.5 2.5 9.25 10

11.         Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 7.5 2.5 0.75 10.5 0.75 5 7.5 2.5 9.25 Area X-coordinate Value (cm) Y-coordinate Value (cm) 7.5 2.5 0.75 10.5 0.75 5 7.5 2.5 9.25 11

12. Q. Determine the centroid of I section beam. I section is symmetric about y axis, so centroid will lie along y axis. x y 12 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 150 15 2.5 75 15 12.5 100 15 22.5 Area X-coordinate Value (cm) Y-coordinate Value (cm) 150 15 2.5 75 15 12.5 100 15 22.5

13. x y 13         Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 150 15 2.5 75 15 12.5 100 15 22.5 Area X-coordinate Value (cm) Y-coordinate Value (cm) 150 15 2.5 75 15 12.5 100 15 22.5

14. Q. Determine the Centroid of Z section beam. 14 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 100 17.5 2.5 37.5 8.75 12.5 25 5 21.25 Area X-coordinate Value (cm) Y-coordinate Value (cm) 100 17.5 2.5 37.5 8.75 12.5 25 5 21.25

15. 15 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 100 17.5 2.5 37.5 8.75 12.5 25 5 21.25 Area X-coordinate Value (cm) Y-coordinate Value (cm) 100 17.5 2.5 37.5 8.75 12.5 25 5 21.25        

16. Moment of force or 1 st Moment of force: Moment of force about any point = product of force and perpendicular distance between the point and line of action of force. 2 nd Moment of force: 1 st moment of force again multiplied by perpendicular distance gives quantity called 2 nd moment of force. It can never be zero due to perpendicular distance between two axis . 1 st Moment of area or 1 st MOI: Area of figure multiplied by perpendicular distance from reference axis is called 1 st moment of area. It is the measure of area distribution of a shape in relation to an axis. Mathematically, Units = (length) 3 2 nd Moment of area/mass or MOI: A rea of figure or mass of body multiplied by square of perpendicular distance from reference axis is called 2 nd moment of area/mass . Moment of Inertia (MOI): 2 nd moment of area/mass is broadly termed as MOI.   16

17. Moment of Inertia (MOI): MOI is always a + ive quantity. It is denoted by I Units = m 4 or mm 4 or (length) 4 Mass moment of inertia: When mass MOI is used in combination with rotation of rigid bodies , it can be considered as “ Measure of Body’s Resistance To Rotation ” . Area moment of inertia: When area MOI is used in combination with deflection or deformation of members in bending, it can be considered as “ Measure of Body’s Resistance To Bending ” . Polar moment of inertia: When polar MOI is used in combination with member subjected to torsional loadings , it can be considered as “ Measure of Body's Resistance To Torsion ” . MOI form the basis of dynamics of rigid bodies and strength of materials. 17

18. MOI of rectangle about centroidal X-axis: MOI of rectangle about centroidal Y-axis:   x x y y C d b MOI changes when axis is changed. In above formula, dimension parallel to centroidal axis will be taken as b (width of beam) and dimensions perpendicular to centroidal axis will be taken as d (depth of beam) . Once b & d are fixed during horizontal axis then never change it, during finding MOI about another vertical axis 18

19. MOI of Hollow Rectangular section: MOI of outer rectangle = MOI of inner rectangle = Similarly ,   When centroid or center of gravity of both rectangles concide each other d d 1 19

20. MOI of circular section: Now MOI about centroidal axis which is perpendicular to circular cross section = MOI of hollow circular cross section:   x x y y C r x x y y C 20

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22. MOI of Composite Figure Moment of inertia of composite section is find out by following steps After drawing axis, split the given composite figure into plane figure (rectangular, circular, triangular). Find the centroid of each plane figure Find the X & Y coordinates of centroid of Composite figure Now use Parallel axis theorem to find MOI about the required axis ( or centroidal axis of Composite figure ).. 22

23. Parallel Axis Theorem MOI of area with respect to any axis in its plane = Centroidal MOI about centroidal axis of small plane figure + the product of area and square of distances between these two parallel axis. Mathematically, MOI w.r.t to X-axis;     MOI w.r.t Y-axis   23

24. If MOI about centroidal axis (of small plane figure) is known, we can find the moment of inertia about any required axis ( or about the centroid of composite figure ) by using Parallel Axis Theorem . From Parallel Axis Theorem, it is clear that MOI increases as reference axis is moved parallel and farther away from centroid. MOI about centroidal axis is least moment of inertia. W hen using Parallel axis theorem, always remember that one of two parallel axis must be centroidal axis. 24

25. MOI of rectangle about centroidal X-axis: MOI of rectangle about centroidal Y-axis:   x x y y C A B C D d b I cx & I cy centroidal MOI or MOI of small figure about its own centroidal axis & = distance between two parallel axis.       MOI w.r.t to X-axis; MOI w.r.t Y-axis   25

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27. Q. Determine the MOI of C section beam w.r.t its centroidal axis. X 3 X 2 X 1 y 3 y 2 y 1 27 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 7.5 2.5 0.75 10.5 0.75 5 7.5 2.5 9.25 Area X-coordinate Value (cm) Y-coordinate Value (cm) 7.5 2.5 0.75 10.5 0.75 5 7.5 2.5 9.25

28.         28 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 7.5 2.5 0.75 10.5 0.75 5 7.5 2.5 9.25 Area X-coordinate Value (cm) Y-coordinate Value (cm) 7.5 2.5 0.75 10.5 0.75 5 7.5 2.5 9.25

29.       b d     29

30.     30 b d      

31. Q. Determine the MOI of I section beam about its centroidal axis I section is symmetric about y axis, so centroid will lie along y axis. x y 31 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 150 15 2.5 75 15 12.5 100 15 22.5 Area X-coordinate Value (cm) Y-coordinate Value (cm) 150 15 2.5 75 15 12.5 100 15 22.5

32. x y 32         Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 150 15 2.5 75 15 12.5 100 15 22.5 Area X-coordinate Value (cm) Y-coordinate Value (cm) 150 15 2.5 75 15 12.5 100 15 22.5 Q. Determine the MOI of I section beam about its centroidal axis

33.     b d     33  

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35. Q. Determine the MOI of Z section beam about its centroidal axis. 35 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 100 17.5 2.5 37.5 8.75 12.5 25 5 21.25 Area X-coordinate Value (cm) Y-coordinate Value (cm) 100 17.5 2.5 37.5 8.75 12.5 25 5 21.25

36. Q. Determine the MOI of Z section beam about its centroidal axis. 36 Area Value X-coordinate Value (cm) Y-coordinate Value (cm) (cm) (cm) 100 17.5 2.5 37.5 8.75 12.5 25 5 21.25 Area X-coordinate Value (cm) Y-coordinate Value (cm) 100 17.5 2.5 37.5 8.75 12.5 25 5 21.25        

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39. Determine the MOI of beam about its centroidal X’ axis. (Hibler page # 789) 39

40. Eg. A-3 , Determine the MOI of beam about its centroidal X & Y axis. (Hibler page # 790) 40

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