   # 6.40 through 6.43 Use the Conjugate-beam method to determine the slopes and deflections at points B and C of the beams Figs. P6.14 through P6.17. FIG. P6.14, P6.40

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Chapter 6, Problem 40P
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## 6.40 through 6.43 Use the Conjugate-beam method to determine the slopes and deflections at points B and C of the beams Figs. P6.14 through P6.17.FIG. P6.14, P6.40 To determine

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

### Explanation of Solution

Calculation:

Consider modulus of elasticity E of the beam is constant.

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

Refer Figure (1),

Consider upward is positive and downward is negative.

Consider clockwise is negative and counterclockwise is positive.

Since the point C is free end there is no support reaction at Point C. Therefore, the reaction at point C is also equal to zero.

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

V=0RAP=0RA=P

Determine the moment at point A;

MA(P×(L2+L2))=0MA=PL

Determine the moment at point B;

MB(P×(L2))=0MB=PL2

Show the M/EI diagram for the real 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).

Calculation of shear at B in the conjugate-beam:

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

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

Determine the shear force at B using the relation;

SB=(bh)(12×b×h)

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

Substitute L2 for b and PL4EI and PL4EI for h.

SB=(L2×PL4EI)(12×L2×PL4EI)=PL28EIPL216EI=2PL2PL216EI=3PL216EI

Hence, the slope at point B is 3PL216EI(Clockwise)_.

The bending moment at B in the conjugate beam is equal to defection at B on the real beam.

Take the clockwise moments of the external forces about B as positive.

Determine the bending moment B using the relation;

MB=(bh)(b2)(12×b×h)(23×b)

Substitute L2 for b and PL4EI and PL4EI for h.

MB=(L2×PL4EI)(L/22)(12×L2×PL4EI)(23×L2)=PL332EIPL348EI=5PL396EI=5PL396EI()

Hence, the deflection at B is 5PL396EI()_

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