Two triangular loads with maximum value $f_0$ and a point force $F$ act on a horizontal beam of length $l$ as depicted above. The triangluar loads reach maximum value at $\frac{l}{3}$ from the left end of the beam.The point force $F$ is applied at the midpoint of the right triangular load $x=\frac{2l}{3}.$ Make a plot of the shear and moment across the beam, marking any points where the shear or moment go to zero.

Identify the equations for shear and moment that you will use to solve

Set up the forces and moments for the solution in vector format

Combine the forces and moments $($if needed$)$ with the relevant principle for static equilibrium $($by vector component$).$

Draw a cross section diagram for each load region.

Solve the equations for shear $V(x)$ and moment $M(x)$ on each cross section diagram in terms of $l,f_0,$ and $F.$

Show by dimensional analysis that $V$ has dimensions of force and $M$ has dimensions of moment.

Insert values $l=10m,f_0=5\frac{N}{m^2},$ and $F=4N$ to make the plots.

Write out the derivation of your solution