A cantilever beam of rectangular cross section is subjected to a load 'W' at its free end .If the depth of the beam is doubled and the load is halved, the deflection of free end as compared to original deflection will be

Concept:

Deflection and slope for a cantilever beam carrying point load at the free end is given by

\(\delta = \frac{{P{\ell ^3}}}{{3EI}},\;\;\theta = \frac{{P{\ell ^2}}}{{2EI}}\)

Calculation:

Given,

bA and bB is width of beam A and B respectively

dA and dB is depth of beam A and B respectively

dB = 2dA , bA = bB and P_B = 0.5PA 

Deflection for a cantilever beam (A) carrying load at free end is given by

\({\rm{\delta_A = }}\frac{{{\rm{P_A}}{{\rm{l}}^{\rm{3}}}}}{{{\rm{3EI_A}}}}\)

where IA is area of moment of inertia of beam A

\({{\rm{I}}_{\rm{A}}} = \frac{{{{\rm{b}}_{\rm{A}}}{\rm{d}}_{\rm{A}}^3}}{{12}}\)

Deflection for a cantilever beam (B) carrying load at free end is given by

\({\rm{\delta_B = }}\frac{{{\rm{P_B}}{{\rm{l}}^{\rm{3}}}}}{{{\rm{3EI_B}}}}\)

where IB is area of moment of inertia of beam A

\({{\rm{I}}_{\rm{B}}} = \frac{{{{\rm{b}}_{\rm{B}}}{\rm{d}}_{\rm{B}}^3}}{{12}}\)

Taking ratio of deflection of beam B to deflection of beam A

\(\frac{{{{\rm{\delta }}_{\rm{B}}}}}{{{{\rm{\delta }}_{\rm{A}}}}} = \frac{{{\rm{P_B}}{{\rm{l}}^3}}}{{3{\rm{E}}{{\rm{I}}_{\rm{B}}}}} \times \frac{{3{\rm{E}}{{\rm{I}}_{\rm{A}}}}}{{{\rm{P_A}}{{\rm{l}}^3}}}\)

\(\frac{{{{\rm{\delta }}_{\rm{B}}}}}{{{{\rm{\delta }}_{\rm{A}}}}} = \frac{\rm{P_B}I_A}{\rm{P_A}I_B}\)

\(\frac{{{{\rm{\delta }}_{\rm{B}}}}}{{{{\rm{\delta }}_{\rm{A}}}}} = \frac{\rm{P_B}b_A{d_A}^3}{\rm{P_A}b_B{d_B}^3}\)

\(\frac{{{{\rm{\delta }}_{\rm{B}}}}}{{{{\rm{\delta }}_{\rm{A}}}}} = \frac{1}{{16}} \)

Hence the deflection of beam B = 6.25 % of deflection of beam A

Imoprtant point:

Deflection calculations using standard results:

	\(\delta = \frac{{W{\ell ^4}}}{{8EI}},\;\theta = \frac{{W{\ell ^3}}}{{6EI}}\)
	\(\delta = \frac{{{\omega _0}{\ell ^4}}}{{30EI}}\;;\theta = \frac{{{\omega _0}{\ell ^3}}}{{24\;EI}}\)
	\(\delta = \frac{{P{\ell ^3}}}{{48EI}}\;;\theta = \frac{{P{\ell ^2}}}{{16EI}}\)
	\(\delta = \frac{5}{{384}}\frac{{\omega {\ell ^4}}}{{EI}}\;\;;\;\;\theta = \frac{{\omega {\ell ^3}}}{{24EI}}\)
	\(\delta = \frac{{{\omega _0}{L^4}}}{{120EI}}\;;\theta = \frac{5}{{192}}\frac{{{\omega _0}{\ell ^3}}}{{EI}}\)
	\({\theta _1} = \frac{{{M _0}\ell }}{{3EI}}\;;{\theta _2} = \frac{{{M _0}\ell }}{{6EI}}\)
	\(\theta = \frac{{{M _0}\ell }}{{4EI}}\)
	\(\delta = \frac{1}{4}\left[ {\frac{{P{\ell ^3}}}{{48EI}}} \right]\)
	\(\delta = \frac{1}{5}\left[ {\frac{5}{{384}}\frac{{{\omega _0}{\ell ^4}}}{{EI}}} \right]\)