Q$\mathrm{.}3$ : Application ofintegralsin structure analyis

$AB$ is a simply umifomly distributed loading beam of w $($kN$/$m$)$ and L span as shown in the

figme below. APply the irtegral to the following:

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

Required:

a$.$ Derive the maximom vabe of $y($deflection$)$ at the middle span of length.

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

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

c$.$ Choose two different vabues of w $(1$&$5$ for one case of span length $(6mm),$sketh

the effect of loading on the vabues of deflection $(y-$rabue$)$ and discuss.

Hint:

EIis constantforbeam cases.

Boundary condions : at $x=0$ & $l,y=0$andatx$=\frac{i}{2}dydx=1$