A cantilever beam of spam 5 m carries two concentrated loads one of 1kN at 3m from the fixed end and the other of 3kN at the free end. If the flexural rigidity EI is 8000 $kN.m^2$. Find the slope and deflection under each load.

Using double integration method,

$BM_x=EI\frac{d^2y}{dx^2}=-3x-1(x-2)$

Integrating,

$EI\frac{dy}{dx}=\frac{-3x^2}{2}-\frac{1(x-2)^2}{2}+c_1$---------(1)

First boundary condition to find $c_1$

$\text{At } x=5; \frac{dy}{dx}=0 \text{ [put in equation (1)]}$

$0=-\frac{3(5)^2}{2}-\frac{1(5-2)^2}{2}+c_1$

$0=-37.5-4.5+c_1$

$c_1=42 \text{ [put in equation (1)]}$

$EI\frac{dy}{dx}=-\frac{3x^2}{2}-\frac{1(x-2)^2}{2}+42$----------------(A)

Integrating again,

$EIy=-\frac{3x^3}{6}-\frac{1(x-2)^3}{6}+42x+c_2$---------(2)

Second boundary condition to find $c_2$

$\text{At x=5 & y=0}$

$0=-\frac{3(5)^3}{6}-\frac{1(5-2)^3}{6}+42\times 5+c_2$

$0=-62.5-4.5+210+c_2$

$c_2=-143 \text{ [put …