   # 6.39 Determine the slope and deflection at point A of the beam shown in Figs. P6.13 and P6.14 by the conjugate-beam method. FIG. P6.13, P6.39

#### Solutions

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Chapter 6, Problem 39P
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## 6.39 Determine the slope and deflection at point A of the beam shown in Figs. P6.13 and P6.14 by the conjugate-beam method. FIG. P6.13, P6.39

To determine

Find the slope and deflection of the beam at point A using conjugate-beam 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 given beam as in Figure (1).

Refer Figure (1),

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)

Since there is no support at point A, there will be no reaction.

Determine the support reaction at B using the relation;

V=0RB12×w×L=0RB=wL2

Consider clockwise moment as negative and counterclockwise moment as positive.

Determine the moment at support B;

+MB=0MB+12×w×L×(23×L)=0MB=wL23

Here, w is the uniformly varying load intensity and L is the distance.

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

Conjugate beam method:

In the real beam A is free end B is a fixed but in conjugate-beam method the end conditions has to be change. Change free end A as fixed end and fixed end B as free.

Show the conjugate beam acted upon by the loading from MEI diagram as in Figure (3).

The slope at A in the real beam is equal to the shear force at A of conjugate beam.

Determine the slope (shear force) at A using the relation;

θA=abwxdx=0Lwx22EI(1x3L)dx

Integrate the equation with respect to x

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