Two cantilever beams are of equal length. One carries a uniformly distributed load and other carries same load but concentrated at the free end. The ratio of maximum deflections is:

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  1. 3/8
  2. 2/3
  3. 1/2
  4. 1/3

Answer (Detailed Solution Below)

Option 1 : 3/8
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Explanation:

Deflection and slope of various beams are given by:

F2 A.M Madhu 09.04.20 D1

 

\({y_B} = \frac{{P{L^3}}}{{3EI}}\)

 

\({\theta _B} = \frac{{P{L^2}}}{{2EI}}\)

F2 A.M Madhu 09.04.20 D2

 \({y_B} = \frac{{w{L^4}}}{{8EI}}\)

 

\({\theta _B} = \frac{{w{L^3}}}{{6EI}}\)

F2 A.M Madhu 09.04.20 D3

 

\({y_B} = \frac{{M{L^2}}}{{2EI}}\)

 

\({\theta _B} = \frac{{ML}}{{EI}}\)

F2 A.M Madhu 09.04.20 D4

 

\({y_B} = \frac{{w{L^4}}}{{30EI}}\)

 

\({\theta _B} = \frac{{w{L^3}}}{{24EI}}\)

F2 A.M Madhu 09.04.20 D5

 

\({y_c} = \frac{{P{L^3}}}{{48EI}}\)

 

\({\theta _B} = \frac{{P{L^2}}}{{16EI\;}}\)


F2 A.M Madhu 09.04.20 D6

 

\({y_c} = \frac{5}{{384}}\frac{{w{L^4}}}{{EI}}\)

 

\({\theta _B} = \frac{{w{L^3}}}{{24EI}}\)


F2 A.M Madhu 09.04.20 D7

 

\({y_c} = 0\)

\({\theta _B} = \frac{{ML}}{{24EI}}\)

F2 A.M Madhu 09.04.20 D8

\({y_c} = \frac{{M{L^2}}}{{8EI}}\)

 

\({\theta _B} = \frac{{ML}}{{2EI}}\)


F2 A.M Madhu 09.04.20 D9

\({y_c} = \frac{{P{L^3}}}{{192EI}}\)

\({\theta _A} = {\theta _B} = {\theta _C} = 0\)

F2 A.M Madhu 09.04.20 D10

\({y_c} = \frac{{w{L^4}}}{{384EI}}\)

\({\theta _A} = {\theta _B} = {\theta _C} = 0\)

Where, y = Deflection of the beam, θ = Slope of beam

From the table,

The maximum deflection of a cantilever beam having uniformly distributed load is given as, 

\({y_A} = \frac{{w{L^4}}}{{8EI}}=\frac{PL^3}{8EI}\)...................(1)

The maximum deflection of a cantilever beam having point load is given as, 

\({y_B} = \frac{{P{L^3}}}{{3EI}}\)...................(2)

From both equation,

\(\frac{y_A}{y_B}=\frac{3}{8}\)

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