A $125$ ton $4-$axle railcar $(75$ in wheel spacing$)$ with self$-$weight of $33$ ton and wheel diameter of $36$ inches. The railcar has a maximum design speed of $45$ mph$.$ The railcar is designed to be used in a mainline with well$-$maintained fir tie. The rail is seated on the timber tie with $24$ ties within $39$ ft rail.

$1)$ Calculate the dynamic loads and select the track modulus? $(2)$

$2)$ Select a rail section and check if the maximum bending stress on this rail section is within allowable range? $(8)$

$3)$ Calculate the maximum rail seat load? $(2)$

$4)$ Select a tie size and check if it meets the requirement of maximum allowable bending moment? $(8)$