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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS A.1 First Moment of An Area; Centroid First Moments of the Area A About the x- and y-Axis are Defined As The centroid of the area A is defined as the point C of coordinates andwhich satisfy the relations

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS The first moment of an area about its symmetric axis is zero, so, the centroid of the area must be on the symmetric axis. When an area possesses a center of symmetry O, the first moment of the area about any axis through O is zero. In other words, O is the centroid of the area.

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS If an area has two symmetric axes, the inter- section of the two axes must be the centroid of the area.

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.1 For the triangular area of Fig. (a), determine (a) the first moment Q x of the area with respect to the x-axis, (b) the coordinate of the centroid of the area. Fig. (a)Fig. (b)

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS dA=udy,dA=udy,

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Complement Problem Determine the first moment of a semi-circular area about the x-axis, (b) the coordinate of the centroid of the semi-circle.

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS A.1 Determination of The First Moment And Centroid of A Composite Area

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.2 Locate the Centroid C of the area A shown in Fig. (a). Fig. (a)Fig. (b)

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Solution: A 1 =80 × 20=1600 mm 2, A 2 =60×40=2400 mm 2

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.3 Referring to the area A of Sample Problem A.2, we consider the horizontal x axis which is through its centroid C. (Such an axis is called a centroidal axis.) Denoting by A the portion of A located above that axis (Fig. a), determine the first moment of A with respect to the x axes. Fig. (a)Fig. (b)Fig. (c)

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Solution: In fact 0

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Complement Problem c (19.7;39.7) x y C1 C2 Determine the centroid of the L-shape area.

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Alternative method: Negative area method x y x

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS A.3 Second Moment, or Moment of Inertia, of An Area; Radius of Gyration Moment of Inertia of A With Respect To the And x Axis And y Axis are Defined, Respectively, As Define the Polar Moment of Inertia of the Area A With Respect To Point O As the Integral :

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Radii of Gyration of An Area A with respect to the x and y axis: Radii of Gyration With Respect To the Origin O

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.4 For the rectangular area of Fig.(a). determine (a) the moment of inertia I x of the area with respect to the centroidal x axis. (b) the corresponding radius of gyration r x. Fig. (a) Fig. (b)

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.5 For the circular area of Fig. (a), determine (a) the polar moment of inertia J O, (b) rectangular moments of inertia I x and I y. Fig. (a)Fig. (b)

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS A.4 Parallel-Axis Theorem

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS A.5 Determination of The Moment of Inertia of a Composite Area.

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.6 Determine the moment of inertia of the area shown with respect to the centroidal x axis (Fig. a). Fig. (a) Fig. (b)

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREASSolution:

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A.6 Product of Inertia for An Area. product of inertia for an element of area located at point (x, y) is defined as If either x or y axis is a symmetric axis, I xy =0.

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Parallel-Axis theorem

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.7 Determine the product of inertia I xy of the triangle shown in Fig. (a).

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.7

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREASSolution:

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Sample Problem A.8 Compute the product of inertia of the beam’s cross- sectional area, shown in Fig. (a), about the x and y centroidal axes.

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREASSolution: Rectangle B Rectangle D The product of inertia for the entire cross section is mm 4 Rectangle A mm 4

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS A.7 Moments of Inertia for An Area About Inclined Axes

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS

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Principal Moments of Inertia

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS

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Sample Problem A.9 Determine the principal moments of inertia for the beam’s cross-sectional area shown in Fig. (a) with respect to an axis passing through the centroid.

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS A.8 Mohr’s Circle for Moments of Inertia

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS Sample Problem A.10 Using Mohr’s circle, determine the principal moments of inertia for the beam’s cross-sectional area, shown in Fig. (a), with respect to an axis passing through the centroid.

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MOMENTS OF AREAS APPENDIX A MOMENTS OF AREAS

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