A cantilever beam $($fixed at its left end at point A$)$ of length L with a point force F at the right end tip C is shown in Figure $1.$ A strain gauge is attached at B with a distance to the fixed end of a$.$ The beam is made of aluminum with the young modulus of E and mass density of rho. It has a rectangular cross$-$section of dimensions b $($width$)$ and h $($height$)$ respectively as shown.

According to the thin beam bending and deflection theory $($Euler$-$Bernoulli beam theory$),$ a generic analytical model should be developed for the beam, to calculate the following design parameters:

Internal bending moment M$($x$)$ and internal shear force f$($x$)$ distributions in the beam along its whole length from the fixed end to the loading end.

The slope function theta$($x$)$ and the deflection function v$($x$)$ of the beam along the whole length of the beam.

Location of the weakest section $($distance to the most left hand of beam$)$ and the weakest point $($point on the weakest section$)$ of the beam.

Parameters:

b$=19$mm

h$=3$mm

L$=380$mm

a$=70$mm

F$=0\mathrm{.}981$N

Youngs modulus E$=69$GPa

Density $=2\mathrm{.}70$g$/$cm$3$