A cantilever is subjected to a uniformly distributed load W(= ω L) over its whole length L, and concentrated upward load W at its free end. The deflection at free end is
Calculation:
Deflection at free end = Deflection due to point load  Deflection due to UDL
\(\delta = \frac{{W{L^3}}}{{3EI}}  \frac{{w{L^4}}}{{8EI}}\)
⇒ \(\delta = \frac{{W{L^3}}}{{3EI}}  \frac{{W{L^3}}}{{8EI}}\) (Where W = wL)
⇒ \(\delta = \frac{{W{L^3}}}{{EI}}\left( {\frac{1}{3}  \frac{1}{8}} \right) = \frac{5}{{24}}\frac{{W{L^3}}}{{EI}}\)
Important Points
Deflection and slope of various beams are given by:
\({y_B} = \frac{{P{L^3}}}{{3EI}}\) 
\({\theta _B} = \frac{{P{L^2}}}{{2EI}}\) 

\({y_B} = \frac{{w{L^4}}}{{8EI}}\) 
\({\theta _B} = \frac{{w{L^3}}}{{6EI}}\) 

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

\({y_B} = \frac{{w{L^4}}}{{30EI}}\) 
\({\theta _B} = \frac{{w{L^3}}}{{24EI}}\) 

\({y_c} = \frac{{P{L^3}}}{{48EI}}\) 
\({\theta _B} = \frac{{w{L^2}}}{{16EI\;}}\) 


\({y_c} = \frac{5}{{384}}\frac{{w{L^4}}}{{EI}}\) 
\({\theta _B} = \frac{{w{L^3}}}{{24EI}}\) 

\({y_c} = 0\) 
\({\theta _B} = \frac{{ML}}{{24EI}}\) 
\({y_c} = \frac{{M{L^2}}}{{8EI}}\) 
\({\theta _B} = \frac{{ML}}{{2EI}}\) 


\({y_c} = \frac{{P{L^3}}}{{192EI}}\) 
\({\theta _A} = {\theta _B} = {\theta _C} = 0\) 
\({y_c} = \frac{{w{L^4}}}{{384EI}}\) 
\({\theta _A} = {\theta _B} = {\theta _C} = 0\) 
Where, y = Deflection of beam, θ = Slope of beam