• Center of gravity

l + a


l + b

„ 3


3 „

Center of gravity

Center of gravity

FIGURE 2.11 Typical M/EI diagram.

diagrams can be broken down into single triangles and rectangles. Beams supporting uniform loads or uniformly varying loads may be handled by integration. Properties of some of the shapes of M/EI diagrams that designers usually come across are given in Figure 2.11.

It should be understood that the slopes and deflections that are obtained using the moment area theorems are with respect to tangents to the elastic curve at the points being considered. The theorems do not directly give the slope or deflection at a point in the beam as compared to the horizontal axis (except in one or two special cases); they give the change in slope of the elastic curve from one point to another or the deflection of the tangent at one point with respect to the tangent at another point. There are some special cases in which beams are subjected to several concentrated loads or the combined action of concentrated and uniformly distributed loads. In such cases it is advisable to separate the concentrated loads and uniformly distributed loads and the moment area method can be applied separately to each of these loads. The final responses are obtained by the principle of superposition.

For example, consider a simply supported beam subjected to a uniformly distributed load q as shown in Figure 2.12. The tangents to the elastic curve at each end of the beam are inclined. The deflection d1 of the tangent at the left end from the tangent at the right end is found to be ql4/24E1. The distance from the original chord between the supports and the tangent at the right end, d2, can be computed as ql4/48E1. The deflection of a tangent at the center from a tangent at the right end, d3, is determined in this step as ql 4/128E1. The difference between d2 and d3 gives the centerline deflection as (5/384)(q(4/EI).

2.2.6 Curved Beams

The beam formulas derived in the previous section are based on the assumption that the member to which a bending moment is applied is initially straight. Many members, however, are curved before a bending moment is applied to them. Such members are called curved beams. In the following discussion all the conditions applicable to the straight-beam formula are assumed valid except that the beam is initially curved.

FIGURE 2.12 Deflection — simply supported beam under uniformly distributed load.

Let the curved beam DOE shown in Figure 2.13 be subjected to the loads Q. The surface in which the fibers do not change in length is called the neutral surface. The total deformations of the fibers between two normal sections such as AB and A1B1 are assumed to vary proportionally with the distances of the fibers from the neutral surface. The top fibers are compressed while those at the bottom are stretched; that is, the plane section before bending remains a plane after bending.

In Figure 2.13 the two lines AB and A1B1 are two normal sections of the beam before the loads are applied. The change in the length of any fiber between these two normal sections after bending is represented by the distance along the fiber between the lines A1B1 and A'B'; the neutral surface is represented by NN1, and the stretch of fiber PP1 is P1P'1, etc. For convenience, it will be assumed that the line AB is a line of symmetry and does not change direction.

The total deformations of the fibers in the curved beam are proportional to the distances of the fibers from the neutral surface. However, the strains of the fibers are not proportional to these distances because the fibers are not of equal length. Within the elastic limit the stress on any fiber in the beam is proportional to the strain in the fiber, and hence the elastic stresses on the fibers of a curved beam are not proportional to the distances of the fibers from the neutral surface. The resisting moment in a curved beam, therefore, is not given by the expression allc. Hence, the neutral axis in a curved beam does not pass through the centroid of the section. The distribution of stress over the section and the relative position of the neutral axis are shown in Figure 2.13b; if the beam were straight, the stress would be zero at the centroidal axis and would vary proportionally with the distance from the centroidal axis as indicated by the dot-dash line in the figure. The stress on a normal section such as AB is called the circumferential stress. Sign Conventions

The bending moment M is positive when it decreases the radius of curvature and negative when it increases the radius of curvature; y is positive when measured toward the convex side of the beam and negative when measured toward the concave side, that is, toward the center of curvature. With these sign conventions, s is positive when it is a tensile stress. Circumferential Stresses

Figure 2.14 shows a free-body diagram of the portion of the body on one side of the section; the equations of equilibrium are applied to the forces acting on this portion. The equations obtained are

Figure 2.15 represents the part ABBjAj of Figure 2.13a enlarged; the angle between the two sections AB and A1B1 is d0. The bending moment causes the plane A1B1 to rotate through an angle Ad0, thereby changing the angle this plane makes with the plane BAC from d0 to (d0 + Ad0); the center of curvature is changed from C to C', and the distance of the centroidal axis from the center of curvature is changed from R to p. It should be noted that y, R, and p at any section are measured from the centroidal axis and not from the neutral axis.

It can be shown that the bending stress s is given by the relation s is the tensile or compressive (circumferential) stress at a point at a distance y from the centroidal axis of a transverse section at which the bending moment is M; R is the distance from the centroidal axis of the section to the center of curvature of the central axis of the unstressed beam; a is the area of the cross-section; Z is a property of the cross-section, the values of which can be obtained from the expressions for various areas given in Table 2.1. (Detailed information can be obtained from Seely and Smith 1952.)


FIGURE 2.14 Free-body diagram of a curved beam segment.
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