Consider the beam represented in Figure $1,$ whose bending stiffness is constant and equal to EI in $AB,CD$ and DE and equal $2EI$ in BC $.$ This beam is subjected to a triangular load in $AB,$a uniformly distributed load in BC and a concentrated load in section D$.$

Figure $1$

$(2)$

$($a$)$ Identify the independent displacements;

$(5)$

$($b$)$ Determine the stiffness matrix, $K,$ and the operator $x_c$ of the complementar solution;

$(4)$

$($c$)$ Determine the fixed$-$end forces vector, $Q_0,$ the nodal forces vector, $Q_N,$ and the operator $x_0$ of the particular solution;

$($d$)$ Solve the equilibrium equation to determine the independent displacements;

$($e$)$ Determine the independent internal forces in all beams;

$($f$)$ Determine the support reactions and represent the internal forces $V$ and $M,$ identifying all the necessary parameters to completely define the corresponding defining functions;

$($g$)$ Qualitatively represent the deformed shape.