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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 1 8.0 SECOND MOMENT OR MOMENT OF INERTIA OF AN AREA 8.1Introduction 8.2Moment of Inertia of an Area 8.3Moment of Inertia of an Area by Integration 8.4Polar Moment of Inertia 8.5Radius of Gyration 8.6Parallel Axis Theorem 8.7 Moments of Inertia of Composite Areas 8.8Product of Inertia 8.9Principal Axes and Principal Moments of Inertia

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 2 8.1. Introduction Forces which are proportional to the area or volume over which they act but also vary linearly with distance from a given axis. -the magnitude of the resultant depends on the first moment of the force distribution with respect to the axis. -The point of application of the resultant depends on the second moment of the distribution with respect to the axis. Herein methods for computing the moments and products of inertia for areas and masses will be presented

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 3 8.2. Moment of Inertia of an Area Consider distributed forceswhose magnitudes are proportional to the elemental areason which they act and also vary linearly with the distance of from a given axis. Example: Consider a beam subjected to pure bending. Internal forces vary linearly with distance from the neutral axis which passes through the section centroid. Example: Consider the net hydrostatic force on a submerged circular gate.

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 4 8.3.Moment of Inertia of an Area by Integration 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 d to be a thin strip parallel to one of the coordinate axes. For a rectangular area, The formula for rectangular areas may also be applied to strips parallel to the axes,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 5 8.4. Polar Moment of Inertia The polar moment of inertia is an important parameter in problems involving torsion of cylindrical shafts and rotations of slabs. The polar moment of inertia is related to the rectangular moments of inertia,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 6 8.5. Radius of Gyration of an Area Consider area A with moment of inertia I x. Imagine that the area is concentrated in a thin strip parallel to the x axis with equivalent I x. k x =radius of gyration with respect to the x axis Similarly,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 7 Examples Determine the moment of inertia of a triangle with respect to its base. SOLUTION: A differential strip parallel to the x axis is chosen for dA. For similar triangles, Integrating dI x from y = 0 to y = h,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 8 a)Determine the centroidal polar moment of inertia of a circular area by direct integration. 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, From symmetry, I x = I y,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 9 8.6. 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. parallel axis theorem

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 10 Moment of inertia I T of a circular area with respect to a tangent to the circle, Moment of inertia of a triangle with respect to a centroidal axis,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 11 8.7. Moments of Inertia of Composite Areas 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 A 1, A 2, A 3,..., with respect to the same axis.

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 12 Example The strength of a W360x57 rolled steel beam is increased by attaching a 225x20 mm plate to its upper flange. Determine the moment of inertia and radius of gyration with respect to an axis which is parallel to the plate and passes through the centroid of the section. SOLUTION: Determine location of the centroid of composite section with respect to a coordinate system with origin at the centroid of the beam section. Apply the parallel axis theorem to determine moments of inertia of beam section and plate with respect to composite section centroidal axis. Calculate the radius of gyration from the moment of inertia of the composite section. 225 mm 358 mm 20 mm 172 mm

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 13 SOLUTION: Determine location of the centroid of composite section with respect to a coordinate system with origin at the centroid of the beam section. 225 mm 358 mm 20 mm 172 mm 189 mm mm 2 mm mm 3 4500 7230 11730 189 850.5 x 10 3

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 14 Apply the parallel axis theorem to determine moments of inertia of beam section and plate with respect to composite section centroidal axis. Calculate the radius of gyration from the moment of inertia of the composite section. 225 mm 358 mm 20 mm 172 mm 189 mm

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 15 Example Determine the moment of inertia of the shaded area with respect to the x axis. SOLUTION: Compute the moments of inertia of the bounding rectangle and half-circle with respect to the x axis. The moment of inertia of the shaded area is obtained by subtracting the moment of inertia of the half-circle from the moment of inertia of the rectangle.

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 16 SOLUTION: Compute the moments of inertia of the bounding rectangle and half-circle with respect to the x axis. Rectangle: Half-circle: moment of inertia with respect to AA’, moment of inertia with respect to x’, moment of inertia with respect to x,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 17 The moment of inertia of the shaded area is obtained by subtracting the moment of inertia of the half-circle from the moment of inertia of the rectangle.

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 18 8.8. Product of Inertia Product of Inertia: When the x axis, the y axis, or both are an axis of symmetry, the product of inertia is zero. Parallel axis theorem for products of inertia:

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 19 8.9. Principal Axes and Principal Moments of Inertia Given we wish to determine moments and product of inertia with respect to new axes x’ and y’. The change of axes yields The equations for I x’ and I x’y’ are the parametric equations for a circle, The equations for I y’ and I x’y’ lead to the same circle. Note:

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 20 At the points A and B, I x’y’ = 0 and I x’ is a maximum and minimum, respectively. I max and I min are the principal moments of inertia of the area about O. The equation for m defines two angles, 90 o apart which correspond to the principal axes of the area about O.

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 21 Example Determine the product of inertia of the right triangle (a) with respect to the x and y axes and (b) with respect to centroidal axes parallel to the x and y axes. SOLUTION: Determine the product of inertia using direct integration with the parallel axis theorem on vertical differential area strips Apply the parallel axis theorem to evaluate the product of inertia with respect to the centroidal axes.

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 22 SOLUTION: Determine the product of inertia using direct integration with the parallel axis theorem on vertical differential area strips Integrating dI x from x = 0 to x = b,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 23 Apply the parallel axis theorem to evaluate the product of inertia with respect to the centroidal axes. With the results from part a,

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 24 For the section shown, the moments of inertia with respect to the x and y axes are I x = 10.38 in 4 and I y = 6.97 in 4. Determine (a) the orientation of the principal axes of the section about O, and (b) the values of the principal moments of inertia about O. SOLUTION: Compute the product of inertia with respect to the xy axes by dividing the section into three rectangles and applying the parallel axis theorem to each. Determine the orientation of the principal axes (Eq. 9.25) and the principal moments of inertia (Eq. 9. 27). Example

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 25 SOLUTION: Compute the product of inertia with respect to the xy axes by dividing the section into three rectangles. Apply the parallel axis theorem to each rectangle, Note that the product of inertia with respect to centroidal axes parallel to the xy axes is zero for each rectangle. 102 mm 76 mm 13 mm 76 mm 31.5 mm 44.5 mm 31.5 mm

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IZMIR INSTITUTE OF TECHNOLOGY Department of Architecture AR231 Fall12/13 18.04.2015 Dr. Engin Aktaş 26 Determine the orientation of the principal axes by the principal moments of inertia by m = 127.5 o m = 37.5 o

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