Task:

Part $1$: Beam analysis

The timber beam in Figure $1$ is simply supported at points A and E $.$ The cross$-$section of the beam changes from segment AC to CE$,$ as shown below. The Young's modulus of timber may be taken as $12$ GPa $.$ The load, geometry and cross$-$sections of the beam will be instructed to each student in class.

$($a$)$ Determine the deflection of the beam at point D by the Virtual Work Method. $[20$ Marks$]$

$($b$)$ Without any calculations, explain the difference if Castigliano's Theorem were uses for $($a$).[5$ Marks$]$

$($c$)$ Calculate the bending and shear stresses at point F on the Section B$-$B$.[15$ Marks$]$

$($d$)$ Sketch the Mohr's circle of stresses corresponding to point F$.[10$ Marks$]$

where;

Load Q$=13\mathrm{.}4$ kn

Length L $=3\mathrm{.}1$ m

Cross$-$section dimensions:

b$1=102$ mm

h$1=221$ mm

b$2=72$ mm

h$2=170$ mm