Presentation on theme: "APPENDIX A MOMENTS OF AREAS"— Presentation transcript:
1APPENDIX A MOMENTS OF AREAS A.1 First Moment of An Area; CentroidFirst Moments of the Area A Aboutthe x- and y-Axis are Defined AsThe centroid of the area A isdefined as the point C of coordinatesandwhich satisfy the relations
2APPENDIX 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.
3APPENDIX 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.
4APPENDIX A MOMENTS OF AREAS Sample Problem A.1For the triangular area of Fig. (a), determine (a) the first moment Qx of the area with respect to the x-axis, (b) the coordinateof the centroid of the area.Fig. (a)Fig. (b)
6APPENDIX A MOMENTS OF AREAS Complement ProblemDetermine the first moment of a semi-circular area about the x-axis, (b) the coordinate of the centroid of the semi-circle.
7APPENDIX A MOMENTS OF AREAS A.1 Determination of The First Moment And Centroid of A Composite Area
8APPENDIX A MOMENTS OF AREAS Sample Problem A.2Locate the Centroid C of the area A shown in Fig. (a).Fig. (a)Fig. (b)
9APPENDIX A MOMENTS OF AREAS Solution:A1=80×20=1600 mm2,A2=60×40=2400 mm2
10APPENDIX A MOMENTS OF AREAS Sample Problem A.3Referring 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)
12c(19.7;39.7) Complement Problem Determine the centroid of the L-shape area.xy10C2c(19.7;39.7)10C180
13Alternative method: Negative area method 8012010xyAlternative method: Negative area methodx
14APPENDIX A MOMENTS OF AREAS A.3 Second Moment, or Moment of Inertia, ofAn Area; Radius of GyrationMoment of Inertia of A With Respect To the And x Axis And y Axis are Defined, Respectively, AsDefine the Polar Moment of Inertia of the Area A With Respect To Point O As the Integral :
15APPENDIX 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
16APPENDIX A MOMENTS OF AREAS Sample Problem A.4For 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)
17APPENDIX A MOMENTS OF AREAS Sample Problem A.5For 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)
18A.4 Parallel-Axis Theorem APPENDIX A MOMENTS OF AREASA.4 Parallel-Axis Theorem
19APPENDIX A MOMENTS OF AREAS A.5 Determination of The Moment of Inertia of a Composite Area.
20APPENDIX A MOMENTS OF AREAS Sample Problem A.6Determine the moment of inertiaof the area shown withrespect to the centroidal x axis (Fig. a).Fig. (a)Fig. (b)
33APPENDIX A MOMENTS OF AREAS Sample Problem A.9Determine 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.
34APPENDIX A MOMENTS OF AREAS A.8 Mohr’s Circle for Moments of Inertia
35APPENDIX A MOMENTS OF AREAS Sample Problem A.10Using 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.