The grid structure has rigid $$supports at points A$,$B and C$.$There is full torsional $($as well as translational and bending$)$restraint at the three supports $($not at D$).$A uniformly distributed load q is applied downwards on element AD and two concentrated loads P are acting downwards at midspan of elements BD and CD$.$The structure has constant modulus of elasticity E and second moment of area I. Axial deformations in the structure can be neglected.

Using the Stiffness Method,

a$.$The numerical values of the components of the $3$x$3$stiffness matrix $[$S$]$corresponding to the three degrees of freedom associated with node D$.$

b$.$The numerical values $($in kN and kNm$)$corresponding to the $3$x$1$forcing vector $\{$Fext$-$F$\}.$

c$.$The numerical values $($in m and radians$)$of the $3$x$1$displacement vector $\{$D$\}.$This calculation involves the inverse of a $3$x$3$stiffness matrix. You can use some supporting software for calculating the inverse of this matrix, i$.$e$.,$Excel or Matlab. However, any other calculation necessary to address both questions of the assignment must be clearly specified, including intermedium steps

d$.$The values $($in kNm$)$at the end nodes and shape of the bending moment diagram for all three beams.

e$.$The values $($in kN and kNm$)$and direction of the reactions $($vertical force and bending and twisting moments$)$at the supports A$,$B and C$.$

P$=30$kN q$=5$kN$/$m L$1=4$m L$2=8$m L$3=9$m