The bending moment at a section is the algebraic sum of the moments about the section of all forces and applied moments on either side of the section.Ī maximum bending moment occurs where the shear or slope of the bending-moment diagram is zero.īending moment is zero where the slope of the elastic curve is at maximum or minimum. The shear at a section is the algebraic sum of all forces on either side of the section. Several important concepts are demonstrated in the preceding examples: From equilibrium conditions for moments about the right end,Īlso, the sum of the vertical forces must equal zero: 3.31 may be determined from the free-body diagram in Fig. For example, the internal forces at the quarter span of the uniformly loaded beam in Fig. It is generally more convenient to use equations of equilibrium to plot the shears, moments, and deflections at critical points along the span. In practice, it is usually not convenient to derive equations for shear and bending-moment diagrams for a particular loading. R1 may then be found from equilibrium of vertical forces:įor bending moment, slope, and deflection can be expressed from x 0 to L/2 and againįor x L/2 to L, as shown in Figs. Summing moments about the left end yields The support reactions R1 and R2 may be determined from equilibrium equations. Similarly, a diagram in which bending moment is plotted along the span is called a bending-moment diagram.Ĭonsider the simply supported beam subjected to a downward-acting, uniformly distributed load w (units of load per unit length) in Fig. A diagram in which shear is plotted along the span is called a shear diagram. The figures also include diagrams indicating the variation of shear, moment, and deformations along the span. Figures 3.28 to 3.49 show some special cases in which shear, moment, and deformation distributions can be expressed in analytic form. However, it may be cumbersome when a large number of concentrated loads act on a structure. For some simple cases this approach can be used conveniently. These relationships suggest that the shear force, bending moment, and beam slope andĭeflection may be obtained by integrating the load distribution. If the slope of the deflected shape is such that d /dxq, the radius of curvature R at a point x along the span is related to the derivatives of the ordinates of the elastic curve (x) by The deflected shape of the beam taken at the neutral axis may be represented by an elastic curve (x). When a beam is subjected to loads, it deflects. To calculate the deflection at various points along a beam, it is necessary to know the relationship between load and the deformed curvature of the beam or between bending moment and this curvature. Resulting internal forces and stresses have been established. To this point, only relationships between the load on a beam and the (3.72) and (3.76) apply to the region of the beam between the concentrated loads.īeam Deflections. When concentrated loads act on a beam, Eqs. This equation indicates that the rate of change in shear at any section equals the load per unit length at that section. Note that when the internal shear acts upward on the left of the section, the shear is positive and when the shear acts upward on the right of the section, it is negative. Figure 3.26b shows the resulting internal forces and moments for the portion of beam dx shown in Fig. A similar relationship exists between the load on a beam and the shear at a section. Indicates that the shear force at a section is the rate of change of the bending moment. The relationship between shear and moment identified in Eq.
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