Question $1$

$(40$ marks$)$

The structure shown in the below figure is fixed at Node A and constrained with roller support at Node $C.$ The structure is subjected to both concentrated vertical load $P=30kN$ and bending moment $M=30kN.m$ at Node B$.$ Assume El is constant.

i$.$ Determine the degree of static indeterminacy $($DSI$),$ i$.$e$.,$ the excess of unknown actions than the available number of equations of static equilibrium.

ii$.$ Find the reaction at support $C$ using the flexibility $($force$)$ method of analysis.

iii. Draw the shear force diagram $($SFD$)$ and bending moment diagram $($BMD$)$ showing all the peak values.

$($Refer to online lecture slides for the integration table if required.$)$

Question $2$

$(60$ marks$)$

For the same structure shown in the figure:

i$.$ Determine the degree of kinematic indeterminacy $($DKI$),$ i$.$e$.,$ the number of independent translations and rotations $($the unknown joint displacements$).$

ii$.$ Develop the stiffness matrix of beam segment $1($connecting A to B$)$ and beam segment $2($connecting $B$ to $C)$ and assemble the global stiffness matrix.

iii. Determine the nodal displacements as well as the element end$-$forces using the stiffness $($displacement$)$ method of analysis.

iv$.$ Draw the shear force diagram $($SFD$)$ and bending moment diagram $($BMD$)$ showing all the peak values.

$($Refer to the online lecture slides for the stiffness matrix of a beam element. You can utilize software such as MS Excel to facilitate matrix computations. You are not required to demonstrate the steps for matrix computations, such as matrix inverse calculations.$)$