   # 6.44 through 6.48 Using the conjugate-beam method, determine the smallest moments of inertia I required for the beams shown in Figs. P6.18 through P6.22, so that the maximum beam deflection does not exceed the limit of 1/360 of the span length (i.e., Δ max ≤ L /360). FIG. P6.18, P6.44

#### Solutions

Chapter
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Chapter 6, Problem 44P
Textbook Problem
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## 6.44 through 6.48 Using the conjugate-beam method, determine the smallest moments of inertia I required for the beams shown in Figs. P6.18 through P6.22, so that the maximum beam deflection does not exceed the limit of 1/360 of the span length (i.e., Δmax ≤ L/360).FIG. P6.18, P6.44 To determine

Find the smallest moment of inertia (I) required for the beam if the deflection does not exceed the limit l/360 of the span length by using conjugate-beam method.

### Explanation of Solution

Given information:

The Young’s modulus is 200 GPa.

Calculation:

Consider the flexural rigidity EI of the beam is constant.

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

Refer Figure (1),

Since support A is a free end there will be no reaction. Therefore, the reaction at support A is automatically zero.

Consider upward is positive and downward is negative.

Consider clockwise is negative and counterclockwise is positive.

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

V=0RC(50+100)=0RC=150kN

Determine the moment at point A;

MA=400kNm

Determine the moment at point B;

MB400(50×6)=0MB=700kNm

Determine the moment at support A;

MA400(50×12)(100×6)=0MA=400+1,200MA=1,600kNm

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

Conjugate-beam method:

In the given beam system, point C is a fixed end and point A 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 A in the conjugate-beam:

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

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

Determine the deflection at A using the relation;

ΔA=ΔAC=[(b1×h1)(b12)(12×b2×h2)(23×b2)(b3×h3)(b1+b32)(12×b4×h4)(23×b1+b4)]

Substitute 6 m for b1, 400EI for h1, 6 m for b2, 300EI for h2, 6 m for b3, 700EI for h3, 6 m for b4, and (1,600EI700EI) for h4

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