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The strength of a W360x57 rolled steel beam is increased by attaching a 225x20 mm plate to its upper flange.
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Apply the parallel axis theorem to determine moments of inertia of beam section and plate with respect to composite section centroidal axis. Engin Aktaşġ2 Example 225 mm 358 mm 20 mm 172 mm SOLUTION: Determine location of the centroid of composite section with respect to a coordinate system with origin at the centroid of the beam section. The moment of inertia of a composite area A about a given axis is obtained by adding the moments of inertia of the component areas A1, A2, A3. Moment of inertia IT of a circular area with respect to a tangent to the circle, Moment of inertia of a triangle with respect to a centroidal axis, Dr. Engin Aktaşġ0 Moment of inertia of a triangle with respect to a centroidal axis, Parallel Axis Theorem Consider moment of inertia I of an area A with respect to the axis AA’ The axis BB’ passes through the area centroid and is called a centroidal axis. b) Using the result of part a, determine the moment of inertia of a circular area with respect to a diameter. SOLUTION: An annular differential area element is chosen, a)ĝetermine the centroidal polar moment of inertia of a circular area by direct integration. Engin AktaşĨ An annular differential area element is chosen, For similar triangles, Determine the moment of inertia of a triangle with respect to its base. Engin Aktaşħ Examples SOLUTION: A differential strip parallel to the x axis is chosen for dA. kx = radius of gyration with respect to the x axis Similarly, Dr. Imagine that the area is concentrated in a thin strip parallel to the x axis with equivalent Ix. Engin AktaşĬonsider area A with moment of inertia Ix. The polar moment of inertia is related to the rectangular moments of inertia, Dr. The polar moment of inertia is an important parameter in problems involving torsion of cylindrical shafts and rotations of slabs. For a rectangular area, The formula for rectangular areas may also be applied to strips parallel to the axes, Dr. Second moments or moments of inertia of an area with respect to the x and y axes, Evaluation of the integrals is simplified by choosing dA to be a thin strip parallel to one of the coordinate axes. Engin AktaşĤ 8.3.Moment of Inertia of an Area by Integration Example: Consider the net hydrostatic force on a submerged circular gate. Consider distributed forces whose magnitudes are proportional to the elemental areas on which they act and also vary linearly with the distance of from a given axis. Internal forces vary linearly with distance from the neutral axis which passes through the section centroid. Engin AktaşĮxample: Consider a beam subjected to pure bending. Herein methods for computing the moments and products of inertia for areas and masses will be presented Dr. The point of application of the resultant depends on the second moment of the distribution with respect to the axis.
the magnitude of the resultant depends on the first moment of the force distribution with respect to the axis.
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Introduction Forces which are proportional to the area or volume over which they act but also vary linearly with distance from a given axis. 1 8.0 SECOND MOMENT OR MOMENT OF INERTIA OF AN AREAĨ.1 Introduction 8.2 Moment of Inertia of an Area 8.3 Moment of Inertia of an Area by Integration 8.4 Polar Moment of Inertia 8.5 Radius of Gyration 8.6 Parallel Axis Theorem 8.7 Moments of Inertia of Composite Areas 8.8 Product of Inertia 8.9 Principal Axes and Principal Moments of Inertia Dr.