What is the lightest I-beam (S-shape section) that will support the loads shown
without exceeding a bending stress of 150 MPa? Base beam selection on those in the appendix. of this eBook.

Solution

Design problems usually cannot be directly solved since the weight of the beam is not known at the start. Generally, the problem is first solved considering only the applied loads. Then, after the beam has been specified for the given loads, the weight of the beam will be added as another load and the problem is solved again.
Sometimes this process takes 2 or 3 iterations but usually, it only takes 1.

Free-Body Diagram
(applied loads only)

The beam needs to withstand the largest possible moment which will occur the wall since this is a cantilever beam.
This moment will be

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