   # Use the conjugate-beam method to determine the slope and deflection at point B of the beams shown in Figs. P6.9 through P6.12. FIG. P6.12, P6.38

#### Solutions

Chapter
Section
Chapter 6, Problem 38P
Textbook Problem
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## Use the conjugate-beam method to determine the slope and deflection at point B of the beams shown in Figs. P6.9 through P6.12. FIG. P6.12, P6.38

To determine

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

### Explanation of Solution

Calculation:

Consider the flexural rigidity EI of the beam is constant.

Show the given beam as in Figure (1).

Consider upward is positive and downward is negative.

Consider clockwise is negative and counterclockwise is positive.

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

MA(w×a×a2)=0MA=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).

Conjugate-beam method:

In the given beam system, point A is a fixed end and point B is free end. But in the conjugate-beam method the fixed end of a real beam becomes free and the free end of real beam changed into the fixed end.

Show the M/EI diagram for the conjugate-beam as in Figure (3).

Consider the external forces acting (left of B) upward on the free body diagram as positive.

The shear force at B of the conjugate beam is equal to the slope at B on the real beam.

Determine the shear force at B using the relation;

SB=(13×b×h)

Here, b is the width of parabolic and h is the intensity of the moment.

Substitute a for b and wa22EI for h.

SB=(13×a×wa22EI)=wa36EI

Hence, the slope at B is wa36EI(Clockwise)_

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