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# ch_3 - 3 Statics of objects in two dimensions 3.1 3.1.1...

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25 3. Statics of objects in two dimensions 3.1 Moment of a 2-D force 3.1.1 Introduction In Chapter 2, we assumed that all forces were concurrent at a single point, a particle and the tendency of a force has been to move the particle in a certain direction. Now we look at the effect of forces on bodies, i.e. we have to take the size of body into consideration. Since forces acting on a body have different points of application, in addition to the tendency to move the body, a force may also tend to rotate a body about an axis. The tendency of a force to rotate is called the moment of a force . Fig. 3.1 Moment of a force. To motivate the idea of a moment we compare the two column-footing designs below. In Fig. 3.1a and Fig. 3.1c, the structure is symmetrical, i.e. the column and footing centerlines are the same. In Fig. 3.1b and Fig. 3.1d, the column and footing centerlines are different, i.e. the column has an eccentricity with respect to the footing. In Fig. 3.1c and Fig. 3.1d we have replaced the two columns by corresponding forces acting on the foundations. In case (c), the force has the tendency to move the foundation downward only. In case (d), the force has the tendency to move the foun- dation downward and rotate the foundation about an axis perpendicular to the plane of the paper. We measure the tendency to make a body rotate about an axis through a point by the moment of the force about that axis. For example, the magnitude of the moment A M of the force F about point A is defined as the product of the magnitude F of the force and the perpendi- cular distance d from A to the line of action of F . A M F d = (3.1) a a a a d a a a a F F d (a) (b) (c) (d) A A

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26 The perpendicular distance is often called moment arm . If we use SI units, the moment has units of Newtonmeters or kilo- newtonmeters (Nm, kNm). In the US system, the moment has units (lb-ft, lb-in, k-ft, k-in). A moment has either a clockwise or a counterclockwise orientation depending on the relative position of the force and the axis. In two-dimensional pro- blems, we consider a counterclockwise moment as positive and a clockwise moment as negative (see Fig. 3.2). It is essential to work sign-consistent within a given problem. Fig. 3.2 Sign of moment. In this class, we restrict our discussion to two-dimensional problems, i.e. we deal with forces that act in a given plane, say the - x y plane. When dealing with coplanar forces, the specification of the moment axis is redundant since all moments are about the - z axis. Thus we commonly speak of the moment about a point. What we are actually implying is a moment about an axis normal to the plane (the - z axis) passing through the point. 3.1.2 Varignon’s theorem The French mathematician Varignon (1654-1722) developed an important theorem of statics, which states: A moment of a force about a point is equal to the sum of the moments of its components about the same point.
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ch_3 - 3 Statics of objects in two dimensions 3.1 3.1.1...

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