Axial Force, Shear, and Bending Moment

Internal forces were defined in Section 3.2 as the forces and couples exerted on a portion of the structure by the rest of the structure. Consider, for example, the simply supported beam shown in Fig. 5.1(a). The freebody diagram of the entire beam is depicted in Fig. 5.1(b), which shows the external loads, as well as the reactions Ax and Ay, and By at supports A and B, respectively. As discussed in Chapter 3, the support reactions can be computed by applying the equations of equilibrium to the free body of the entire beam. In order to determine the internal forces acting on the cross section of the beam at a point C, we pass an imaginary section cc through C, thereby cutting the beam into two parts, AC and CB, as shown in Figs. 5.1(c) and 5.1(d). The free-body diagram of the portion AC (Fig. 5.1(c)) shows, in addition to the external loads and support reactions acting on the portion AC, the internal forces, Q; S, and M exerted upon portion AC at C by the removed portion of the structure. Note that without these internal forces, portion AC is not in equilibrium. Also, under a general coplanar system of external loads and reactions, three internal forces (two perpendicular force components and a couple) are necessary at a section to maintain a portion of the beam in equilibrium. The two internal force components are usually oriented in the direction of, and perpendicular to, the centroidal axis of the beam at the section under consideration, as shown in Fig. 5.1(c). The internal force Q in the direction of the centroidal axis of the beam is called the axial force, and the internal force S in the direction perpendicular to the centroidal axis is referred to as the shear force (or, simply, shear). The moment M of the internal couple is termed the bending moment. Recall from mechanics of materials that these internal forces, Q; S, and M, represent the resultants of the stress distribution acting on the cross section of the beam.


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