Find the moment of inertia about an axis 10.0 cm to the left of the y axis. Eg) The following shape has a centroidal area moment of inertia of I y = 600.0 cm4 and an area of 25.0cm2. Note: the bar over the I is to differentiate the centroidal moment of inertia with moment of inertia about another axis. If we are interested in the moment of inertia about axis other than the centroidal axis x, the following formula is used: I x I x Ad 2 and I y I y Ad 2 where I x is the moment of inertia about the centroid, A is the area of the shape and d the distance between the two axis. Notice that the table in appendix C only provides the formula for I through a centroidal axis( x ). y x d x In the above example, we are asked to find the area moment of inertia about axis x. Parallel Axis Thereom The centroidal moment of inertia can be used to find the value of moments of inertia about other parallel axes. From appendix C the formulas for Ix and Iy of a quarter circle is I I x x and I y r 4 16 (150mm) 16 4 9.940x107 mm4 For the half circle: (9.940x107mm4) x 2 = 1.99 x 108 mm4 Ix and Iy = 1.99 x 108 mm4 Eg) Calculate I x for the following area: y(mm) 60 25 x(mm) 60 For the circle: I x 4r I x 4 (25mm) 4 4 3.068x105 mm4 For the rectangle: b I x 12h Ix 3 (60 mm )( 60 mm )3 1.08 x10 6 mm 4 12 For the total area: I x = 1.08 x 106mm4 - 3.068 x 105 mm4 = 7.73 x 105 mm4 Notice that area moment of inertia of the circle is subtracted from the area moment of inertia of the rectangle because the circle is cutout. Eg) Calculate Ix and Iy for the following area: y(mm) 150 x(mm) The area is composed of two quarter circles. This causes the shape to have a greater resistance to bending around the x axis and therefore a larger moment of inertia around that axis. Eg) Determine I x and I y for the following area: y 90cm X 60cm From Appendix C of the textbook, the Ix and Iy of a rectangle with the x and y axis through its centroid are: b I x 12h 3 3 h I b12 and y 3 I (60cm) (90cm) x y (60cm) (90cm) 1.62x10 cm 12 3.65x106 cm4 3 I 6 4 12 It is logical that Ix is greater than Iy because a larger amount of the rectangular area lies further from the x axis than the y axis. I 1 14123 - 1 10 83 12 12 I = 2016mm4 - 427mm4 I = 1589mm4 Notice that the volumes of the two beams are approximately the same, but because Beam 2 has a larger area moment of inertia it will resist bending more than Beam 1. For Beam 1 the moment of inertia is : I 1 10 83 12 I = 427mm4 For Beam 2, the moment of inertia is: I for the entire rectangle subtract I for the cutout. Eg) Determine the Area Moment of Inertias for the following two beams: 8mm 8mm 10mm 10mm 2mm Beam 1 Beam 2 From Appendix C of the textbook, the moment of inertia about an x-axis through a rectangle’s centroidal axis is: I 1 3 bh 12 Note: b is always denoted as being the side parallel to the x-axis. Area moment of inertias for common areas are found in any statics textbook. Different objects have different moment of inertia formula depending on the shape and the axis to which the moment is been taken about. From these formula, the specific moment of inertia formula are determined using calculus. The above formula are the moment of inertia formula for objects in general. In general the formula for the area moment of inertia for an object is: Ix = Ay2 Iy = Ax2 The units of area moment of inertia are m4, mm4 etc. The area moment of inertia is the summation of all the individual areas that make up the object times the square of the distance each area is from an given axis. F x F x A 2 x 4 on edge has more area of the board, a greater distance from the x-axis and therefore has a larger area moment of inertia. A 2 x 4 will have a greater resistance to bending if put on its side then if laid flat. A beam with a larger moment of inertia will resist bending more than a beam with a smaller area moment of inertia. Īlso note that unlike the second moment of area, the product of inertia may take negative values.Area Moment of Inertia The area moment of inertia of an object is a measure of the resistance that object gives to bending. Principal axes Reference Table Area Moments of Inertia
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