A new high$-$capacity class $10$ combine is under production and will utilize up to a $61$ foot wide header. For such a wide header to work, the unloading auger must extend outward beyond the edge of the header to allow unloading into a grain cart while harvesting. The current design uses a $24-$foot long $24-$inch diameter tube as the main unloading auger housing. This housing contains a $22-$inch diameter auger with $\frac{1}{8}-$inch steel flighting on a $12-$inch pitch. The center shaft of the auger is built from a $2\mathrm{.}5-$inch pipe weighing $13l\frac{b}{f}t.$ The new design will require extending the unloading auger housing to a total length of $30$ feet. Refer to this information and the images below as you answer the following questions. State all assumptions in each problem set.$[$Stress$/$Deflection$]$ Assume the manufacturer plans to purchase cold formed steel tubing conforming to ASTM A$500$ with a tensile strength of $58$ ksi and yield strength of $46$ ksi $.$ Consider the weight of the housing and auger. Estimate the volume of the auger flight by equating to a rectangular cross$-$section extruded along a helix on the outer surface of the main shaft $(2\mathrm{.}5-$inch pipe$).$ Use $0\mathrm{.}286l\frac{b}{i}{n}^3$ for the density of steel.

$[$Fatigue Failure$]$ Assume the end of the auger housing deflects $+-25\%$ both forwardly and downwardly of the original position while in operation due to rough conditions in the field. This deflection increases or reduces the normal bending stress due to the weight as it fluctuates. Hint: Back calculate the load using the max deflection formula