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2.2.2 Shear Force and Bending Moment Diagrams

To plot the shear force and bending moment diagrams it is necessary to adopt a sign convention for these responses. A shear force is considered to be positive if it produces a clockwise moment about a point in the free body on which it acts. A negative shear force produces a counterclockwise moment about the point. The bending moment is taken as positive if it causes compression in the upper fibers of the beam and tension in the lower fiber. In other words, a sagging moment is positive and a hogging moment negative. The construction of these diagrams is explained with an example given in Figure 2.4.

The section at E of the beam is in equilibrium under the action of applied loads and internal forces acting at E as shown in Figure 2.5. There must be an internal vertical force and internal bending moment to maintain equilibrium at section E. The vertical force or the moment can be obtained as the algebraic sum of all forces or the algebraic sum of the moment of all forces that lie on either side of the section E.

The shear on a cross-section — an infinitesimal distance to the right of point A is +55 and therefore the shear diagram rises abruptly from 0 to +55 at this point. In the portion AC, since there is no additional load, the shear remains at +55 on any cross-section throughout this interval, and the diagram is a horizontal as shown in Figure 2.4. At an infinitesimal distance to the left of C the shear is +55, but at an infinitesimal distance to the right of this point the concentrated load of magnitude 30 has caused the shear to be reduced to +25. Therefore, at point C, there is an abrupt change in the shear force from +55 to +25. In the same manner, the shear force diagram for the portion CD of the beam remains a rectangle. In the portion DE, the shear on any cross-section a distance x from point D is

S = 55 - 30 - 4x = 25 - 4x which indicates that the shear diagram in this portion is a straight line decreasing from an ordinate of +25 at D to +1 at E. The remainder of the shear force diagram can easily be verified in the same way. It should be noted that, in effect, a concentrated load is assumed to be applied at a point, and hence at such a point the ordinate to the shear diagram changes abruptly by an amount equal to the load.

55 k 55

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