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**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 and which satisfy the relations

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**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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**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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**APPENDIX A MOMENTS OF AREAS**

Sample Problem A.1 For the triangular area of Fig. (a), determine (a) the first moment Qx 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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**APPENDIX A MOMENTS OF AREAS**

dA=udy,

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**APPENDIX A MOMENTS OF AREAS**

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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**APPENDIX A MOMENTS OF AREAS**

A.1 Determination of The First Moment And Centroid of A Composite Area

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**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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**APPENDIX A MOMENTS OF AREAS**

Solution: A1=80×20=1600 mm2, A2=60×40=2400 mm2

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**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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**APPENDIX A MOMENTS OF AREAS**

Solution: In fact

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**c(19.7;39.7) Complement Problem**

Determine the centroid of the L-shape area. x y 10 C2 c(19.7;39.7) 10 C1 80

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**Alternative method: Negative area method**

80 120 10 x y Alternative method: Negative area method x

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**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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**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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**APPENDIX A MOMENTS OF AREAS**

Sample Problem A.4 For the rectangular area of Fig.(a). determine (a) the moment of inertia Ix of the area with respect to the centroidal x axis. (b) the corresponding radius of gyration rx. Fig. (b) Fig. (a)

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**APPENDIX A MOMENTS OF AREAS**

Sample Problem A.5 For the circular area of Fig. (a), determine (a) the polar moment of inertia JO, (b) rectangular moments of inertia Ix and Iy. Fig. (a) Fig. (b)

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**A.4 Parallel-Axis Theorem**

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

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**APPENDIX A MOMENTS OF AREAS**

A.5 Determination of The Moment of Inertia of a Composite Area.

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**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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**APPENDIX A MOMENTS OF AREAS**

Solution:

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**APPENDIX A MOMENTS OF AREAS**

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, Ixy=0.

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**Parallel-Axis theorem**

APPENDIX A MOMENTS OF AREAS Parallel-Axis theorem

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**APPENDIX A MOMENTS OF AREAS**

Sample Problem A.7 Determine the product of inertia Ixy of the triangle shown in Fig. (a).

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**APPENDIX A MOMENTS OF AREAS**

Sample Problem A.7

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**APPENDIX A MOMENTS OF AREAS**

Solution:

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**APPENDIX A MOMENTS OF AREAS**

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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**APPENDIX A MOMENTS OF AREAS**

Solution: Rectangle A mm4 Rectangle B Rectangle D mm4 mm4 The product of inertia for the entire cross section is mm4

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**APPENDIX A MOMENTS OF AREAS**

A.7 Moments of Inertia for An Area About Inclined Axes

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**APPENDIX A MOMENTS OF AREAS**

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**APPENDIX A MOMENTS OF AREAS**

Principal Moments of Inertia

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**APPENDIX A MOMENTS OF AREAS**

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**APPENDIX A MOMENTS OF AREAS**

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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**APPENDIX A MOMENTS OF AREAS**

A.8 Mohr’s Circle for Moments of Inertia

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**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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**APPENDIX A MOMENTS OF AREAS**

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