Equation 4.3 suggests the following expression:Įquation 4.4 states that the change in the shear force is equal to the area under the load diagram. Similarly, the shearing force at section x + dx is as follows:Įquation 4.3 implies that the first derivative of the shearing force with respect to the distance is equal to the intensity of the distributed load. Equation 4.1 suggests the following expression:Įquation 4.2 states that the change in moment equals the area under the shear diagram. The equation also suggests that the slope of the moment diagram at a particular point is equal to the shear force at that same point. To compute the bending moment at section x + dx, use the following:Įquation 4.1 implies that the first derivative of the bending moment with respect to the distance is equal to the shearing force. The total load acting through the center of the infinitesimal length is wdx. Let the shear force and bending moment at a section located at a distance of x from the left support be V and M, respectively, and at a section x + dx be V + dV and M + dM, respectively. Sign conventions for axial force, shearing force, and bending moment.Ĥ.4 Relation Among Distributed Load, Shearing Force, and Bending Momentįor the derivation of the relations among w, V, and M, consider a simply supported beam subjected to a uniformly distributed load throughout its length, as shown in Figure 4.3. If the bending moment tends to cause concavity downward (hogging), it will be considered a negative bending moment (see Figure 4.2e and Figure 4.2f). Similarly, a shear force that has the tendency to move the left side of the section downward or the right side upward will be considered a negative shear force (see Figure 4.2c and Figure 4.2d).Ī bending moment is considered positive if it tends to cause concavity upward (sagging). Such force is regarded as compressive, while the member is said to be in axial compression (see Figure 4.2a and Figure 4.2b).Ī shear force that tends to move the left of the section upward or the right side of the section downward will be regarded as positive. On the other hand, an axial force is considered negative if it tends to crush the member at the section being considered. Such a force is regarded as tensile, while the member is said to be subjected to axial tension. As a convention, the positive bending moments are drawn above the x-centroidal axis of the structure, while the negative bending moments are drawn below the axis.Īn axial force is regarded as positive if it tends to tier the member at the section under consideration. This is a graphical representation of the variation of the bending moment on a segment or the entire length of a beam or frame. As a convention, the shearing force diagram can be drawn above or below the x-centroidal axis of the structure, but it must be indicated if it is a positive or negative shear force. This is a graphical representation of the variation of the shearing force on a portion or the entire length of a beam or frame. The bending moment (BM) is defined as the algebraic sum of all the forces’ moments acting on either side of the section of a beam or a frame. The phrase “on either side” is important, as it implies that at any particular instance the shearing force can be obtained by summing up the transverse forces on the left side of the section or on the right side of the section. The shearing force (SF) is defined as the algebraic sum of all the transverse forces acting on either side of the section of a beam or a frame. The normal force at any section of a structure is defined as the algebraic sum of the axial forces acting on either side of the section. In this chapter, the student will learn how to determine the magnitude of the shearing force and bending moment at any section of a beam or frame and how to present the computed values in a graphical form, which is referred to as the “shearing force” and the “bending moment diagrams.” Bending moment and shearing force diagrams aid immeasurably during design, as they show the maximum bending moments and shearing forces needed for sizing structural members. To predict the behavior of structures, the magnitudes of these forces must be known. When a beam or frame is subjected to transverse loadings, the three possible internal forces that are developed are the normal or axial force, the shearing force, and the bending moment, as shown in section k of the cantilever of Figure 4.1.
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