(Solved): A railroad tie (or sleeper) is subjected to two rail loads, each of magnitude P=200kN, acting as s ...

A railroad tie (or sleeper) is subjected to two rail loads, each of magnitude $P=200\mathrm{kN}$, acting as shown in the figure. The reaction $q$ of the ballast is assumed to be uniformly distributed over the length of the tie, which has cross-sectional dimensions $b=280\mathrm{mm}$ and $h=150\mathrm{mm}$. Calculate the maximum bending stress in the tie due to the loads $P$, assuming $L=2,000\mathrm{mm}$ and $a=500\mathrm{mm}$.