Subject : Structural Analysis 1

Topic : Deflection of beam

Difficulty : High

Using the Double Integration Method,

$ BM_x=EI\frac{d^2y}{dx^2}=-w\times x\times\frac{x}{2}=-\frac{wx^2}{2}$

Integrating, $EI\frac{dy}{dx}=-\frac{wx^3}{6}+c_1$ ----- (1)

First boundary condition: at $x=L,\ \frac{dy}{dx}=0$

From equation (1)

$ 0=\frac{-wL^3}{6}+c_1\implies c_1=\frac{wL^3}{6}\\ \boxed{c_1=\frac{wL^3}{6}}\ [put\ in\ equation\ (1)]$

$ EI\frac{dy}{dx}=-\frac{wx^3}{6}+\frac{wL^3}{6}\dots\dots\dots[G.S.E]$ ----- (A)

Integrating again,

$EIy=\frac{wx^4}{24}+\frac{wL^3}{6}x+c_2$ ------ (2)

Second boundary condition at x=L, y=0

$\therefore from\ equation\ (2)\\ 0=\frac{-wL^4}{24}+\frac{wl^4}{6}+c_2\\ \therefore …