## 821

Spans AC and BC FIGURE 2.10 Example — continuous beam.

where M is the bending moment at the point and EI is the flexural rigidity of the beam section. Since the deflection is small, 1/R is approximately taken as and Equation 2.13 may be rewritten as

In Equation 2.14, y is the deflection of the beam at distance x measured from the origin of the coordinate. The change in slope in a distance dx can be expressed as M dx/EI and hence the slope in a beam is obtained as r M

Equation 2.15 may be stated as follows: the change in slope between the tangents to the elastic curve at two points is equal to the area of the M/EI diagram between the two points.

Once the change in slope between the tangents to the elastic curve is determined, the deflection can be obtained by integrating further the slope equation. In a distance dx the neutral axis changes in direction by an amount d0. The deflection of one point on the beam with respect to the tangent at another point due to this angle change is dd = x d0, where x is the distance from the point at which deflection is desired to the particular differential distance.

To determine the total deflection from the tangent at one point A to the tangent at another point B on the beam, it is necessary to obtain a summation of the products of each d0 angle (from A to B) times the distance to the point where deflection is desired or dB - dA =

1 Mx dx EI

The deflection of a tangent to the elastic curve of a beam with respect to a tangent at another point is equal to the moment of M/EI diagram between the two points, taken about the point at which deflection is desired.

### 2.2.5.1 Moment Area Method

The moment area method is most conveniently used for determining slopes and deflections for beams in which the direction of the tangent to the elastic curve at one or more points is known, such as cantilever beams, where the tangent at the fixed end does not change in slope. The method is applied easily to beams loaded with concentrated loads, because the moment diagrams consist of straight lines. These

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