Chapter 5 DISTRIBUTED FORCES: CENTROIDS AND CENTERS OF GRAVITY The center of gravity of a rigid body is the point G where a single force W, called the weight of the body, can be applied to represent the effect of the earth’s attraction on the body. Consider two-dimensional bodies, such as flat plates and wires contained in the xy plane. By adding forces in the vertical zdirection and moments about the horizontal y and xaxes, the following relations were derived W = dWxW = x dWyW = y dW which define the weight of the body and the coordinates x and y of its center of gravity.
xA = x dAyA = y dA For a homogeneous flat plate of uniform thickness, the center of gravity G of the plate coincides with the centroid C of the area A of the plate, the coordinates of which are defined by the relations These integrals are referred to as the first moments of area Awith respect to theyand x axes, and are denoted by Qy and Qx , respectively Qy = xA Qx = yA
Similarly, the determination of the center of gravity of a homogeneous wire of uniform cross section contained in a plane reduces to the determination of the centroid C of the line L representing the wire; we have xL = x dLyL = y dL y C x L y O x
X Y X SW = S xW Y SW = S yW W3 z y y z W1 W2 SW G3 G G2 O O G1 x x The areas and the centroids of various common shapes are tabulated. When a flat plate can be divided into several of these shapes, the coordinates X and Y of its center of gravity G can be determined from the coordinates x1, x2. . . and y1, y2. . . of the centers of gravity of the various parts using
y z G O x If the plate is homogeneous and of uniform thickness, its center of gravity coincides with the centroid C of the area of the plate, and the first moments of the composite area are Qy = X SA = S xA Qx = Y SA = S yA
When the area is bounded by analytical curves, the coordinates of its centroid can be determined by integration. This can be done by evaluating either double integrals or a single integral which uses a thin rectangular or pie-shaped element of area. Denoting by xeland yel the coordinates of the centroid of the element dA, we have Qy = xA = xel dAQx = yA = yel dA
L The theorems of Pappus-Guldinus relate the determination of the area of a surface of revolution or the volume of a body of revolution to the determination of the centroid of the generating curve or area. The area A of the surface generated by rotating a curve of length L about a fixed axis is C y x 2py A = 2pyL where y represents the distance from the centroid C of the curve to the fixed axis.
C A y x 2py The volume V of the body generated by rotating an areaA about a fixed axis is V = 2pyA where y represents the distance from the centroid C of the area to the fixed axis.
dW W w x w W = A w C x B O B x O P dx x L L The concept of centroid of an area can also be used to solve problems other than those dealing with the weight of flat plates. For example, to determine the reactions at the supports of a beam, we replace a distributed load wby a concentrated load W equal in magnitude to the area A under the load curve and passing through the centroid C of that area. The same approach can be used to determine the resultant of the hydrostatic forces exerted on a rectangular plate submerged in a liquid.
The coordinates of the center of gravity Gof a three- dimensional body are determined from xW = xdWyW = y dWzW = z dW For a homogeneous body, the center of gravity G coincides with the centroidCof the volume V of the body; the coordinates of C are defined by the relations xV = xdVyV = y dVzV = z dV If the volume possesses a plane of symmetry, its centroid C will lie in that plane; if it possesses two planes of symmetry, C will be located on the line of intersection of the two planes; if it possesses three planes of symmetry which intersect at only one point, C will coincide with that point.
The volumes and centroids of various common three- dimensionalshapes are tabulated. When a body can be divided into several of these shapes, the coordinates X, Y, Zof its center of gravity G can be determined from the corresponding coordinates of the centers of gravity of the various parts, by using X SW = S xW Y SW = S yW Z SW = S zW If the body is made of a homogeneous material, its center of gravity coincides with the centroid Cof its volume, and we write X SV = S xV Y SV = S yV Z SV = S zV
1 2 zel z xel = x When a volume is bounded by analytical surfaces, the coordinates of its centroid can be determined by integration. To avoid the computation of triple integrals, we can use elements of volume in the shape of thin filaments (as shown). P(x,y,z) yel = y zel = z z dV = z dx dy y xel dx yel x dy Denoting by xel, yel , and zel the coordinates of the centroid of the element dV, we write xV = xel dVyV = yel dVzV = zel dV If the volume possesses two planes of symmetry, its centroid C is located on their line of intersection.
xel y xel= x dV= pr 2dx x z dx If the volume possesses two planes of symmetry, its centroid C is located on their line of intersection. Choosing the x axis to lie along that line and dividing the volume into thin slabs parallel to the xz plane, the centroid C can be determined from xV = xel dV For a body of revolution, these slabs are circular.