Beam:

Material: Structural ASTM$-$A$36$Steel $($E $=200$GPa$)$

Length: Lb $=10$m

profile is W$310$x$310$x$342$

Duct:

Material: Aluminum $($E $=69$Gpa$,$and density $2700$kg$/$m$3)$

Length: La $=8$m

Profile: Hallow tube of $50$mm thickness, and outer diameter given.

Hangers:

Material: Cable $-$stainless AISI $302($E $=180$GPa$)$and diameter $50$mm

Spring $($K $=200$kN$/$m$)$

Length: h $=10$m

Number of hangers: $6$

Spacing: b $=1$m

Outer diameter$=250$mm

Considering the self$-$weights of both the duct and the beam, perfom the following tasks to analyse the problem:

Task $1$: Symbolic solution framework $($Beam and duct treated individually$)$ Setup the symbolic solution frameworks for determining shear force, bending moment, slope and deflection in the beam and duct. Follow the problem solving steps outlined below. NOTE: Not all steps presented below may be needed.

$1.$ Load analysis $-$ Free Body Diagram $($FBD$)$

a$.$ Self$-$weight $-$ Write the expression for self$-$weight distribution $\backslash ($ w$\_\{$o$\}\backslash )($per unit length$),$ given material density and profile dimension.

b$.$ External $-$ Write the equilibrium equations and if possible, solve for reactions

c$.$ Internal $-$ Write the expression for the internal shear force $\backslash ($ V$($x$)\backslash ),$ and bending moment $\backslash ($ M$($x$)\backslash )-$ Macaulay equations

$2.$ Material behaviour $-$ Obtain the equations for slope $\backslash (\backslash$theta$($x$)\backslash )$ and deflection $\backslash ($ v$($x$)\backslash )-$integrate the moment equation.

$3.$ Geometry of deformation $-$ Define boundary conditions $-$ known deflections and slopes at specific points $-$ solve for integration constants where possible.

Task $2$: Symbolic relation of hangers to the deformations of the beam and duct Update the boundary conditions using the following cases of hanger types and obtain equations for slope and deflection in each case:

$1.$ Rigid hangers

$2.$ Deformable cables

$3.$ Springs

Task $3$: Matlab program

Given a set of numberical information of the system $($uploaded on SunLearn$),$ write a matlab program to plot the shear force, bending moment, slope and deflection diagrams in the beam and the duct.

Task $4$: Results analysis with design improvements

Compile your solution report answering specific quastions as asked on the report template uploaded on sunlearn. Submit your report in PDF format by Friday, $\backslash (\backslash$mathbf$\{27\}\backslash )$ September, $17$h$00,$ on SunLearn