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MOMENTS OF INERTIA MRS SITI KAMARIAH BINTI MD SAAT SCHOOL OF BIOPROCESS ENGINEERING UNIVERSITI MALAYSIA PERLIS

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At the end of this topics, student should able to: 1. Develop method for determining the moment of inertia for an area 2. Determine the mass moment of inertia.

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Or called second moment of area, I Measures the efficiency of that shape its resistance to bending Moment of inertia about the x-x axis and y-y axis.

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Unit : m 4, mm 4 or cm 4

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xx b h y y

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Rectangle at one edge I uu = bh 3 /3 I vv = hb 3 /3 Triangle I xx = bh 3 /36 I nn = hb 3 /6 xx v v h b b h u u nn

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I xx = I yy = d 4 /64 I xx = (BH 3 -bh 3 )/12 I yy = (HB 3 -hb 3 )/12 xx y y B H b h xx y y

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Used to find the moment of inertia of an area about centroidal axis.

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Calculate the moment of inertia at z-z axis. b=150mm;h=100mm; d=50mm d zz xx

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Calculate the moment of inertia about x-x axis 400 mm 24 mm 12 mm d= 212 mm xx 200 mm

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I xx of web = (12 x 400 3 )/12= 64 x 10 6 mm 4 I xx of flange = (200 x 24 3 )/12= 0.23 x 10 6 mm 4 I xx from principle axes xx = 0.23 x10 6 + Ad 2 Ad 2 = 200 x 24 x 212 2 = 215.7 x 10 6 mm 4 I xx from x-x axis = 216 x 10 6 mm 4 Total I xx = (64 + 2 x 216) x10 6 =496 x 10 6 mm 4

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Unit of length Used in design of columns in structure.

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Many cross-sectional areas consist of a series of connected simpler shapes, such as rectangles, triangles, and semicircles. In order to properly determine the moment of inertia of such an area about a specified axis, it is first necessary to divide the area into its composite parts and indicate the perpendicular distance from the axis to the parallel centroidal axis for each part. Use the moment of inertia of an area or parallel axis theorem.

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1. Subdivide the cross-section into three- part A,B,D 2. Determine moment of inertia of each part, for rectangular, I=bh 3 /12. 3. Use the parallel axis theorem formula for each part. 4. Summation for entire cross-section.

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Try Fundamental problems Problems: 10-49 till 10-56.

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To use this method, first determine the product of inertia for the area as well as its moments of inertia for given x, y axes.

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Units: m 4, mm 4. Product of inertia may either +ve, -ve or zero depending on the location and orientation of the coordinate axes. If the axis symmetry for an area, product of inertia will be zero.

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Passing through the centroid of the area.

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Determine the product of inertia about the x and y centroid

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1. Subdivide the cross-section into three-part A,B,D 2. Determine product moment of inertia

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Total up the product moment of inertia

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Try Example 10.7 Problems: 10-71,10-75-78,10-82

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A measure of the bodys resistance to angular acceleration. Used in dynamics part, to study rotational motion. Mass moment of inertia of the body: Where r= perpendicular distance from the axis to the arbitrary element dm.

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The axis that is generally chosen for analysis, passes through the body s mass center G If the body consists of material having a variable density = (x, y, z), the element mass dm of the body may be expressed as dm = dV Using volume element for integration, When being a constant,

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Shell Element For a shell element having height z, radius y and thickness dy, volume dV = (2 y)(z)dy Disk Element For disk element having radius y, thickness dz, volume dV = ( y 2 ) dz

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For moment of inertia about the z axis, I = I G + md 2

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For moment of inertia expressed using k, radius of gyration,

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Try: Problems: 10-89 -95,96,97,99,100

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1. The definition of the Moment of Inertia for an area involves an integral of the 2. SI units for the Moment of Inertia for an area. 3. The parallel-axis theorem for an area is applied to . 4. The formula definition of the mass moment of inertia about an axis is

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5. Calculate the moment of inertia of the rectangle about the x-axis 2cm 3cm x

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GOOD LUCK !!!