ERT 146 ENGINEERING MECHANICS MOMENTS OF INERTIA MRS SITI KAMARIAH BINTI MD SA’AT SCHOOL OF BIOPROCESS ENGINEERING UNIVERSITI MALAYSIA PERLIS

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

Moment of Inertia 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.

Moment of Inertia of an Area Unit : m4, mm4 or cm4

Moment of Inertia for common solid shapes x b h y

Moment of Inertia of common shapes v b h h x x u u n n v b Rectangle at one edge Iuu= bh3/3 Ivv= hb3/3 Triangle Ixx= bh3/36 Inn= hb3/6

Moment of Inertia of common shapes y B y h x H x x x b y y Ixx = (BH3-bh3)/12 Iyy = (HB3-hb3)/12 Ixx= Iyy = πd4/64

Parallel-Axis Theorem for an Area Used to find the moment of inertia of an area about centroidal axis.

Example 10-1: Calculate the moment of inertia at z-z axis. b=150mm;h=100mm; d=50mm d z x

Example 10-2: Calculate the moment of inertia about x-x axis x x 400 mm 24 mm 12 mm d= 212 mm 200 mm

Solution: Ixx of web = (12 x 4003)/12= 64 x 106 mm4 Ixx of flange = (200 x 243)/12= 0.23 x 106 mm4 Ixx from principle axes xx = 0.23 x106 + Ad2 Ad2 = 200 x 24 x 2122 = 215.7 x 106 mm4 Ixx from x-x axis = 216 x 106 mm4 Total Ixx = (64 + 2 x 216) x106 =496 x 106 mm4

Radius of Gyration of an Area Unit of length Used in design of columns in structure.

Moment of Inertia for Composite Areas 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.

Example 10-3 (Example 10.5):

Solution Subdivide the cross-section into three-part A,B,D Determine moment of inertia of each part, for rectangular, I=bh3/12. Use the parallel axis theorem formula for each part. Summation for entire cross-section.

Moment of Inertia for Composite Areas Try Fundamental problems Problems: 10-49 till 10-56.

Product of Inertia for an Area 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.

Product of Inertia for an Area Units: m4, mm4. 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.

Parallel-Axis Theorem Passing through the centroid of the area.

Example 10-3 Determine the product of inertia about the x and y centroid

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

Solution Total up the product moment of inertia

Product of Inertia for an Area Try Example 10.7 Problems: 10-71,10-75-78,10-82

Mass Moment of Inertia A measure of the body’s 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.

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 ρ = ρ(x, y, z), the element mass dm of the body may be expressed as dm = ρ dV Using volume element for integration, When ρ being a constant,

Procedure for Analysis Shell Element For a shell element having height z, radius y and thickness dy, volume dV = (2πy)(z)dy Disk Element For disk element having radius y, thickness dz, volume dV = (πy2) dz

Parallel-Axis Theorem For moment of inertia about the z axis, I = IG + md2

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

Example 10-4

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

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

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

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