QUESTION $1(40\%)$

The elements $AB,BC,BE,DE$ and EF of the frame structure in Figure $1$ have negligible axial deformation, and bending stiffness equal to $EI($modulus of elasticity multiplied by second moment of area$).$ The element BF has negligible bending stiffness, and axial stiffness equal to $EA($or modulus of elasticity multiplied by cross$-$sectional area$).$ The structure is subjected to a uniformly distributed load $q$ applied on beam BC and an imposed vertical settlement $$ at support $D.$ Using the stiffness method based on the four independent degrees of freedom shown in Figure $1(D_1=$ rotation of node E; $D_2=$ vertical displacement of node E; $D_3=$ horizontal displacement of node $B$;$D_4=$ rotation of node B $),$ you are asked to:

Q$1($a$).$ Provide the stiffness matrix $[$S$]$ associated with the four degrees of freedom $($vector $\{D\}.$ Note that in the stiffness method, $[S]\{D\}=\{F_{\text{e}\text{x}\text{t}}-F\}.$

$Q1(b).$ Provide the forcing vector $\{F_{\text{e}\text{x}\text{t}}-F\}.$

Note: The components of your stiffness matrix and forcing vector solutions must be provided as a function of $a,b,c,E($modulus of elasticity$),A($cross$-$sectional area$),($imposed settlement at support $D)$ and $q.$