   # 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. P6.21, P6.47

#### Solutions

Chapter
Section
Chapter 6, Problem 47P
Textbook Problem
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## 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. P6.21, P6.47

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 30 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),

Consider upward forces are positive and downward forces are negative.

Consider clockwise moments are negative and counterclockwise moments are positive.

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

MD=0RA×15(120×10)(120×3)=0RA=1,56015RA=104kN

Determine the reaction at support C;

V=0RA+RD120120=0RD=240104RD=136kN

Show the reactions of the given beam as in Figure (2).

Refer Figure (2),

Determine the bending moment at support A;

MA=(136×15)(120×12)(120×5)=2,0402,040=0

Determine the bending moment at B;

MB=(104×5)=520kNm

Determine the bending moment at C;

MC=(136×3)=408kNm

Determine the bending moment at D;

MD=+(104×15)(120×10)(120×3)=2,0402,040=0

Show the M/EI diagram of the given beam as in Figure (3).

Show the conjugate-beam as in Figure (4).

Determine the support reaction at A of a conjugate MEI diagram using the relation;

MD=0RA(15)+(12(520EI)(5)(13×5+10)(7)(408EI)(3+72)12(7)(520EI408EI)(23×7+3)12(3)(408EI)(23×3))=0RA=1EI(15,166.67+18,564+3,005.33+1,224)15RA=2,530.67kNm2EI

The maximum deflection will occur for the given beam at point M and take a distance of xM from point B.

Determine the intensity of load at xM using the relation;

wMxM=112kNm7mwM=1127xM

Determine the shear force at M using the relation;

SM=[RA+(12×b1×h1)+(xm×h2)12×wM×xM]=0

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

Substitute 2,530.67EI for RA, 5 ft for b1, 520EI for h1, 520EI for h2, and 1127xM for wM.

[2,530.67EI+(12×5×520EI)+(xm×520EI)12×(1127xM)×xM]=01EI(2,530

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