Q$\mathrm{.}1$: Taking nodes at A$,$ B$,$ C$,$ and D of the beam shown below, identify the nodal displacementsQ$\mathrm{.}1$: Taking nodes at A$,$ B$,$ C$,$ and D of the beam shown below, identify the nodal displacements

$($DOF$)$ of the structure and write the nodal displacement vector $\{x\}$ in terms of these DOF.

Calculate the nodal load vector $\{P\}$ and the stiffness matrix $K$ for the beam.

Solve the governing equation $[K]\{x\}=\{P\}$ to calculate the nodal displacement vector $\{x\}.$

Using the values of $\{x\},$ determine the member end forces and moments, and combine those with

the corresponding contributions due to fixed end conditions.

Utilise these values, draw the shear force $($SF$)$ and bending moment $($BM$)$ diagrams of the beam.

Show the values of SF and BM at important sections $($member end sections, sections subjected

point loads, maximum moment with a member subjected to distributed load$).$ Also show the sign

conventions you used for the SF and BM$.$

The beam has uniform flexural rigidity $(EI)$ where $E=200$GPa and $I=50\mathrm{.}0{10}^{6}m{m}^4.$

Note: the placement of the load $5kN$ on $AB$ is unsymmetrical $-$ calculate its fixed end forces

carefully using moment equilibrium equation taking its free body diagram

As the $4kN$ force is acting directly at node $A($nodal load$)$ and not inside the member $AB($not

member load$),$ it will not contribute any fixed end moment $-$ thus, it need not be considered

unnecessarily in the calculations of fixed end forces and moments. However, it must be considered

finally in the calculation of $\{P\}$ where it will simply be added algebraically $($considering proper

direction$/$sign$)$ as the nodal load.

$($DOF$)$ of the structure and write the nodal displacement vector $\{$X$\}$ in terms of these DOF.

Calculate the nodal load vector $\{$P$\}$ and the stiffness matrix $[$K$]$ for the beam.

Solve the governing equation $[$K$]\{$X$\}=\{$P$\}$ to calculate the nodal displacement vector $\{$X$\}.$

Using the values of $\{$X$\},$ determine the member end forces and moments, and combine those with

the corresponding contributions due to fixed end conditions.

Utilise these values, draw the shear force $($SF$)$ and bending moment $($BM$)$ diagrams of the beam.

Show the values of SF and BM at important sections $($member end sections, sections subjected

point loads, maximum moment with a member subjected to distributed load$).$ Also show the sign

conventions you used for the SF and BM$.$

The beam has uniform flexural rigidity $($EI$)$ where E $=200$GPa and I $=50\mathrm{.}0$x$106$ mm$4$