   # Determine the slope and deflection at point B of the beam shown by the moment-area method. FIG. P6.12, P6.38

#### Solutions

Chapter
Section
Chapter 6, Problem 12P
Textbook Problem
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## Determine the slope and deflection at point B of the beam shown by the moment-area method. FIG. P6.12, P6.38

To determine

Find the slope θB and deflection ΔB at point B of the given beam using the moment-area method.

### Explanation of Solution

Calculation:

Consider the flexural rigidity EI of the beam is constant.

Show the given beam as in Figure (1).

Consider clockwise as negative and counterclockwise as positive.

Determine the moment at support A using the Equation of equilibrium;

MA=w×a×a2=wa22

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

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

Elastic curve:

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

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

The slope at point B 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:

θB=θBA=AreaoftheM/EIbetweenAandB=12×b×h

Here, b is the width and h is the height of the triangle.

Substitute a for b and wa22EI for h.

θB=θBA=13×a×wa22EI=wa36EI

Hence, the slope at point B is wa36EI_

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