5.1 PREFACE
5.2 CENTRE OF MASS
5,3 CENTROID
5.4 CENTRES OF MASS AND CENTROIDS OF COMPOSITE BODIES
5.5 THEOREM OF PAPPUS
5.6 FIRST MOMENT OF AREA
5.7 SECOND MOMENT OF AREA
5.8 SECOND MOMENT OF AREA FOR COMPOSITE AREAS
5.9 SUMMARY
Notes
5.1 PREFACE
In Chapter 2, Section 2.15, we had seen forces that are not concentrated at their lines of action and points of application. The forces are distributed over their fields of action. In real situations, all forces are distributed over an area bigger than a point (represented by their point of application). However, in previous chapters of this book, all forces had been assumed to be concentrated on their points of action. In those cases, the assumption yields answers acceptable for engineering analyses because the areas of action of the forces do not significantly influence the result of the analyses and the external effects f the forces n the bodies they acted upon are close to the effects of the corresponding concentrated forces.
When a force acts over an area whose size influences the result of analyses significantly, the force is called a distributed force. It can no longer be handled as a concentrated force. Analyses need to be done through methods specially developed for it.
A distributed force has a resultant that acts at point or centre of distribution that depends on the intensity of the force and its nature of distribution (i.e. how the intensity changes from one point to another over the field of action of the force). For example, the gravitational force that acts on a body has a resultant, viz. the weight of the body, that acts at its centre of gravity f the body. Hence, the study on distributed forces shall be started, in this chapter, by by focusing our attention of the concept of the centre of gravity. For practical purposes in engineering, the study covers the centre of mass of the body as well. After that, we shall look at two geometrical properties of figures required in distributed force analyses, namely the centroid and the second moments of area of a body. The analysis on the effects of the actions of distributed forces shall subsequently be studied in Chapter 6.
5.2 CENTRE OF MASS
The force of gravitational pull acting on a body has been portrayed through the weight W of the body which acts at the centre of gravity G of the body. The position of the centre of gravity of a body is determined, mathematically, by applying the Varignon theorem (or the principle of moment) to the gravitational forces acting on the elements of the body.
Consider the body in Figure 5 whose position is described by the right-hand coordinate system (x,y,z).
5,3 CENTROID
5.4 CENTRES OF MASS AND CENTROIDS OF COMPOSITE BODIES
5.5 THEOREM OF PAPPUS
5.6 FIRST MOMENT OF AREA
5.7 SECOND MOMENT OF AREA
5.8 SECOND MOMENT OF AREA FOR COMPOSITE AREAS
5.9 SUMMARY
The chapter describes geometrical properties of lines, areas, and volumes. The early part of the chapter describes the centroid for lines, two-dimensional bodies, and three-dimensional bodies. The method …..
The following section discusses two properties important to areas, namely the first moment and second moment (also called the moment of inertia) of areas. The study on the second moment of areas begins with first principle analyses, i.e. through the integration of differential elements. Methods of choosing the appropriate differential elements were explained. Subsequently, we discussed the method for calculating the second moments for composite bodies through direct summation of the values of the second moments of areas of their component bodies. In the discussions, we were also introduced to the parallel axes principle and the concept of radius of gyration.
The materials studied in this chapter are, as a matter of fact, tools for the study on the action of distributed loads. The principles discussed will be used in the analysis of distributed forces to be discussed in the next chapter.
Notes
Note (1): Guidelines For Defining The Differential Element To Locate The Centroid.
When applying Equation (5.4) and Equation (5.5) to determine the centroid of a volume or an area, we must first specify the differential element to be integrated. The element chosen will determine how easy or difficult is the integration process that you need to be carry out to solve the expression produced from Equation (5.4) and Equation (5.5). In forming the differential element, there are a number of guidelines to be followed so that the expressions obtained are easy to solve. The five guidelines are spelled-out below:
Guideline 1. Coordinate System. The coordinate system chosen must be the most suitable for the figure to be analysed. For example, a figure with a circular boundary is normally easily to handle by using the polar coordinate system.
Guideline 2. Element (Order and Continuity):
Guideline 2(a): Order Of The Differential Elements. As far as possible, use a differential element with the lowest order . Try to use the first order element. i.e. an element with only one differential size (the sizes in the other directions being finite). A first order element needs one single integration to cover the whole body. For example, a general element for an area, Figure 5C.1, is taken as dA= x dy. The element is a strip with a finite length x and only one differential size dy, Figure 5C1(b). The solution is obtained through a single integration w.r.t. y. If an element with two differential sizes (dA = dx dy) is used, we need two integrations to solve (one with respect of x and another with respect to y. For a volume, Figure 5C.2, we normally use an element represented by:
- a disc, Figure 5C.2(a), dV= phi y^2 dz
- a shell, Figure 5C.2(b), dV = 2 phi yh dh.
The disc and the shell both require one integration to solve. On the other hand, to solve the element dV = h dx dy [shown in Figure 5C.2(c)] requires two integrations and the element dV = dx dy dz [shown in Figure 5C.2(d)] requires three integrations.
Guideline 2(b): Continuity of the expressions. Note the continuous property of an element. As far as possible, use an element that can cover the whole figure through a single integration expression. For example, the element in Figure 5C.3(a) needs one integration only whilst the element in Figure 5C.3(b) requires two integrations due to the discontinuity in the expression for length y of the element at the position x=x<0>.
Guideline 3. Moment Arm For The Element. When taking the moments for a differential element of first or second order, the moment arm used is the distance of the centroid from the reference axis. This distance may or may not be the same as the distance from the boundary of the element to the axes. For example, for the element in Figure 5C.4, the distance uscd is x<C> and not the boundary distance x when the moment is taken from the y-axis. However, for the moment from the x-axis, the difference between the boundary distance y and the centroidal distance y<C> can be neglected.
Guideline 4. Higher Order Expressions. The expression for higher orders in an expression can be neglected. Only the expression of the lowest order is considered. For example, the shaded areas (1/2) dx dy for the element in Figure 5C.5 can be neglected in comparison to the area y dx.
Guideline 5. Symmetry. Area A (or line L) is said to be symmetrical about any axis a-a if, for every point P on the area (or line), there exist a corresponding point P’, with the line PP’ being perpendicular to and bisected by the axis a-a , Figure 5C.6. Note that:
- If an area A (or line L) possesses an axis of symmetry, its centroid is located on the axis of symmetry.
- If the axis of symmetry is taken as the reference axis for measuring the location of the centroid, the coordinate of the centroid in the direction perpendicular to the axis, is zero.
- If a figure possesses two axes of symmetry, its centroid coincides with the intersection of the axes of symmetry (for example, Figure 5C.7).
Note (2): Practical Applications Of Second Moments Of Areas.
Note (3): Defining An Element For Integrating When Finding The Second Moment Of Area.
Note (4): Choosing the Coordinate System For Measuring The Second Moment Of Area.
Engineering Mechanics - Statics 