A cantilever beam of length, L, with uniform cross-section and flexural rigidity, EI, is loaded uniformly by a vertical load, w per unit length. The maximum vertical deflection of the beam is given by

Concept:

Area moment method

Theorem 1: The difference of slope of any two points of a beam is equal to the area of\(\frac{M}{{EI}}\) diagram between those points.

θ_B – θ_A = Area of \(\frac{M}{{EI}}\) diagram between B and A.

Theorem 2: The difference of deflection of two points of a beam is equal to the moment of area of\(\frac{M}{{EI}}\)diagram between those points.

Y_B – Y_A = Moment of area of\(\frac{M}{{EI}}\)diagram between B and A.

Y_B – Y_A = (Ax?)

Calculation:

δ_max = Ax? 

\({δ _{max}} = \frac{1}{3} \times L \times \frac{{W{L^2}}}{{2EI}} \times \frac{3}{4}L\)

\(\Rightarrow {δ _{max}} = \frac{{W{L^4}}}{{8EI}}\)