A bridge is modelled by a $4$th order differential equation. The bridge made up of steel, on simple supports.

The loading has three parts.

$,$ Its own weight, $W$

An upward wind force acting under the bridge expressed by

$F=-\frac{1}{\sqrt[\phantom{2}]{2}}{x}^{4\mathrm{.}5}{e}^{-\frac{1}{2}(\frac{x-}{}{)}^2},=1,=150$

Equal magnitude upward point forces due to hangers, evenly distributed, each as W$/10.$

The modulus of elasticity $E=200$GPa

The density is $=7850k\frac{g}{m^3}$

$g=9\mathrm{.}81\frac{m}{s^2}$

$a=200m$

$b=20m$

$h=4m$

$L=1000m$

$I=\frac{1}{12}b{h}^3$

The beam differential equation is

$EI{w}^{iv}-P{w}^{\text{'}\text{'}}=F$

$EI{w}^{iv}-P{w}^{\text{'}\text{'}}=0$

and there are $4$ BC$\text{'}$s $($simply supported$).$

$F$ is the vertical external force including the weight to be calculated from the given data. $P$ is the axial tensile force acting on the beam $($suppose there is so$)P=\frac{W}{5}$

Solve the differential equation $($find the deflection of the beam$)$ using Green's function for

a$)$ axial force case

b$)$ no axial force case.

Hint:

Use the Green's function for the weight

Use the Green's function for the wind lift force

Use the Green's function for the $(4)$ point hanger forces separately.

Use superposition principle due to linearity of differential equation

Express the effects of all forces including the weight, separately $0$

Express the total effect of all forces including the weight

Find the maximum displacement, slope, bending moment and shear force, and their locations in the bridge

Send the computer codes.