   # 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. P6.9, P6.35

#### Solutions

Chapter
Section
Chapter 6, Problem 35P
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. P6.9, P6.35

To determine

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

### Explanation of Solution

Given information:

The Young’s modulus (E) is 200 GPa.

The moment of inertia (I) is 800×106mm4

Calculation:

Consider 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 bending moment at point A using the relation;

MA(P×l)=0

Here, P is the load and l is the distance.

Substitute 90 kN for P and 5 m for l.

MA(90×5)=0MA=450kNm

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 (2).

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 using the relation;

SB=(12×b×h)

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

Substitute 5.0 m for b and 450EI for h.

SB=(12×5×(450EI))=1,125kNm2EI

Determine the slope at point B using the relation;

θB=1,125kNm2EI

Substitute 200 GPa for E and 800×106mm4 for I.

θB=1,125kNm2(200GPa×106kN/m21GPa)(800×106mm4×(1m1,000mm)4)=0

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