Chapter 5: Distributed Forces; Centroids and Centers of ...

1 Chapter 5: Distributed Forces; Centroids and Centers of GravityForces that act on a body per unit length, area or volume. They are notdiscrete forces that act at specific points. Rather they act over a continuous region. What are Distributed forces? Center of GravityGravity pulls each and every particle of a body vertically is the location of the equivalent single forcethat replaces all the Distributed the location bexyzWednesday, October 07, 200912:39 PM CE297-FA09-Ch5 Page 1 Find the Center of Gravity of the area shown , October 13, 20097:07 PM CE297-FA09-Ch5 Page 2 Centroid of a Quarter or Semi :Wednesday, October 14, 20093:21 PM CE297-FA09-Ch5 Page 3 Centroids of LinesxzyxzyExercise the Centroid of the wire , October 16, 200910:38 AM CE297-FA09-Ch5 Page 4 CE297-FA09-Ch5 Page 5 Centroids and First Moments of Areas & first moment of an area with respect to a line of symmetry is zero.

2 If an area possesses a line of symmetry, its centroid lies on that axis If an area possesses two lines of symmetry, its centroid lies at their intersection. An area is symmetric with respect to an axis BB if for every point Pthere exists a point P such that PP is perpendicular to BB and is divided into two equal parts by BB . The centroid of the area coincides with the center of symmetry. An area is symmetric with respect to a center Oif for every element dAat (x,y) there exists an area dA of equal area at (-x,-y). Properties of SymmetryCentroid of any area always exists. But, a center of symmetrymay or may notexist. NOTE:First Moment of an AreaDefinition: First Moment of a LineDefinition: Wednesday, October 14, 200912:13 PM CE297-FA09-Ch5 Page 6 Composite Areas and LinesThe Centroid of an area (or line) that is made up of several simple shapes can be found easily using the Centroids of the individual an area is composed by adding some shapes and subtracting other shapes, then the momentsof the subtracted shapes need to be subtractedas :Friday, October 16, 20097:53 AM CE297-FA09-Ch5 Page 7 Exercise the centroid of the figure the reactions at A & B.

3 (specific weight = lb/in 3; thickness t= 1 in)Exercise uniform circular rod of weight 8 lb and radius r=10 in is the tension in the cable AB & the reaction at , October 16, 20099:04 AM CE297-FA09-Ch5 Page 8 Surfaces & Volumes of Revolution: Theorems of Pappus-GuldinusThe concepts of area, Centers of areas, and Moments of areas can also be extended to general 3D same integral formulas still hold:General 3D surfaces(aside)Surfaces of revolution are obtained when one "sweeps" a 2-D curve about a fixed of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid through the 1 SurfaceareasofrevolutionRotating abouty-axis:Rotating about x-axis:Sunday, October 18, 20093:55 PM CE297-FA09-Ch5 Page 9 Volume of a body of revolution is equal to the generating area times the distance traveled by the centroid through the 2 Rotating abouty-axis:Rotating about x-axis.

4 Volumes of RevolutionExercise the internal surface area and the volume of the punch R= 250 , October 18, 20098:00 PM CE297-FA09-Ch5 Page 10 Distributed Loads on Beamsw(x)is weight per unit several applications, engineers have to design beams that carry Distributed loads along their supported BeamTotal weight:Point of action:Exercise the reactions at the , October 21, 20099:33 AM CE297-FA09-Ch5 Page 11 Distributed forces on submerged that are submerged in water (or in any liquid) are subjected to Distributed force per unit areawhich is called pressure. In water, this pressure always acts perpendicular (normal) to the submerged surfaceand its magnitude is given by:Aside: If the liquid is viscous, then in addition the normal pressure the viscous fluid may also apply a tangential tractionto the body. This traction is also a force per unit area and is a more general form of buoyancy forceis the resultantof all these Distributed forces acting on the body.

5 Recall the buoyancy force is equal to the weight of the water :Resultant forceTo obtain the resultant force acting on a submerged surface:For inclined surfaces:Exercise x 4m wall of the tank is hinged at A and held by rod BC. Find the tension in the rod as a function of the water depth , October 23, 20097:51 AM CE297-FA09-Ch5 Page 12 For Curved Surfaces:Forces on curved submerged surfaces can he obtained by using equilibrium of a surrounding portion of , October 23, 20099:42 AM CE297-FA09-Ch5 Page 13 CE297-FA09-Ch5 Page 14 Center of Gravity in 3D space; Center of volumeThe formulas for center of gravity in 2 D can be easily generalized to 3D as Composition of VolumesExamples and in the , October 23, 200910:15 AM CE297-FA09-Ch5 Page 15 Center of Volume by complex 3D shapes, triple integrals can be difficult to evaluate exactly.(i) Bodies of revolution(ii) Volume under a surfaceFor some special cases one can find the centroid as follows:Read Example the centroid of the volume obtained by rotating the shaded area about the , October 26, 200911:04 AM CE297-FA09-Ch5 Page 16