Q$\mathrm{.}3$: Application of integrals in structure analysis

$AB$ is a simply uniformly distributed loading beam of $w(k\frac{N}{m})$ and $L$ span as shown in the

figure below. Apply the integral to the following:

$EI\frac{d^{2}y}{d{x}^2}=\frac{wl}{2}x-\frac{w{x}^2}{2}$

Required:

a$.$ Derive the maximum value of $y($deflection$)$ at the middle span of length.

b$.$ Choose different values of span L $($varied from $3-7$m$)$ for one case of loading

$(w=5k\frac{N}{m}),$ sketch a graph showing the effect of span length on deflection $(y$ value$)$

c$.$ Choose two different values of $w(1$&$5)k\frac{N}{m}$ for one case of span length $(6m),$ sketch

the effect of loading on the values of deflection $(y-$value$)$ and discuss.

Hint:

EI is constant for beam cases.

Boundary conditions: at $x=0$ & $l,y=0$ and at $x=\frac{l}{2}$;$d\frac{y}{d}x=0$