   # Determine the slope and deflection at point A of the beam shown by the moment-area method. FIG. P6.13, P6.39

#### Solutions

Chapter
Section
Chapter 6, Problem 13P
Textbook Problem
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## Determine the slope and deflection at point A of the beam shown by the moment-area method. FIG. P6.13, P6.39

To determine

Find the equations for slope and deflection of the beam at point A using moment area-method.

### Explanation of Solution

Calculation:

Consider flexural rigidity EI of the beam is constant.

Consider a section at a distance of x from the free end.

Show the section and the free body diagram of the beam as in Figure (1).

Refer Figure (1),

Consider clockwise as negative and counterclockwise as positive.

Determine the moment at support B;

MB=12×w×L×(23×L)=wL23

Here, w is the uniformly distributed load and L is the distance.

Show the section and M/EI diagram for the given beam as in Figure (2).

Determine the intensity of load at x;

wL=wxxwx=wLx

Determine the bending moment at distance x;

Mx=12×wLx×x(L13×x)=12wx2(1x3L)

Elastic curve:

The sign of M/EI diagram is negative, therefore, the beam bends downward. The support B of the given beam is fixed and the slope at B is zero. Therefore, the tangent to the elastic curve at B is horizontal.

Show the elastic curve diagram as in Figure (3).

The slope at point A can be calculated by evaluating the change in slope between A and B.

Express the change in slope using the first moment-area theorem as follows:

θA=θAB=AreaoftheM/EIbetweenAandB=0LEEIdx=0Lwx22EI(1x3L)dx        (1)

Integrate Equation (1) with respect to x.

θA=θAB=w2EI0L(x2x33L)dx=w2EI(x33x43L×4)0L=w2EI(L33L4

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