Stucture design -I (Centre of Gravity ;Moment of Inertia)

Lecture &
Centre of Gravity
Moment of Inertia

1. Structure I Lecture 7 Centre of gravity (C.G.) & Moment of Inertia (MOI)

2. Centre of gravity (C.G.) The centre of gravity of a body is that point in space through which the resultant pull of the earth , i.e. the ‘weight’ of the body , acts, for all possible positions of the body. If a body is broken in small parts, the pull of gravity on each part will form a parallel force system the magnitude of which will be the sum of the weight of parts.

3. Centroid The point at which the total area of a plane figure (lamina) is assumed to be concentrated is known as centroid of that area. A plane figure like rectangle, triangle, quadrilateral, circle etc having an area but negligible thickness is known as Lamina.

4. Difference between CG and centroid CG deals with solid bodies having mass(in which whole mass is assumed to be concentrated at a specific point) while centroid is used for plane figure like rectangle, circle, triangle etc having negligible thickness(in which whole area is assumed to be concentrated at a specific point).

5. Centroid of some plane figures

6. CG of some solid bodies

7. Steps for locating the centroid of a lamina Choose X-axis through the lower most point/edge of the lamina and Y-axis through left most point/edge of the lamina, locate origin at the intersections of chosen axis and mark the axis and origin on the figure. Divide the lamina into regular elements areas like rectangle, square, circle, semicircle etc. Calculate the following for each elemental area: a i : Area of elemental area In case any elements area is removed from the main lamina, treat that as negative area. x i : Distance of the centroid of the elements from the chosen Y- axis y i : Distance of the centoid of the elements from chosen X-axis

8. Contd. Calculate a i .x i and a i .y i for element area. Complete the following centroid table with relevent units in each column. Sr. No. Elemental Area a i (units) x i (units) y i (units) a i .x i (units) a i .y i (units)

9. Contd. Calculate- Σ a i = Algebraic sum of the areas of all the elemental areas. Σ a i .x i = Algebraic sum of the product ai and xi Σ a i .y i = Algebraic sum of the product ai and yi Calculate x¯ and y¯ from the following equation: x¯ = Σ ai.xi / Σ ai y¯ = Σ ai.yi / Σ ai

10. MOMENT OF INERTIA Inertia is the resistance of any physical object to a change in its state of motion or rest. It is represented numerically by an object's mass. Moment of Inertia is also known as the second moment of a force or an area. (the first moment is the product of force and the arm)

11. The product of the area and the square of the distance of the centroid of the area from an axis is known a moment of inertia of the area about that axis. MOI of an area about X-axis is written as I xx and about Y-axis as I yy . MOI about an axis passing through the centroid of lamina is represented by I G

12. MOI of some common plane geometrical figures

13. PARALLEL AXIS THEOREM This theorem states that “ The Moment of Inertia of a lamina about any axis in the plane of the lamina equals the sum of the following: Moment of Inertia about a centroidal axis parallel to the axis about which moment of inertia is to be calculated Product of the area of the lamina and the square of the distance between the above centroidal axis and the axis about which the moment of inertia is to be calculated.” b d X X’ b d X I about XX’ = bd³/12 I about XX’’ = I XX’ + Ay² = bd³/12 + ( bd ×(d/2)²)

14. THEOREM OF PERPENDICULAR AXIS This theorem states that the MOI of any figure about the axis perpendicular to two mutually perpendicular axes is equal to the sum of MOI about these two axes of the same figure. mathematically, Izz = Ixx + Iyy Z-axis is called polar axis and Izz is known as Polar moment of inertia.