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.

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.

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

     M_w = M_max = Fd + 2(9)
           = 0.5(5)(4) (4/3 + 1) + 18
           = 23.33 + 18 = 41.33 kN-m

The Section Modulus will be

      S_min = M_w / σ_allow
              = 41.33 / 150,000
        = 275.5×10^-6 m³
        = 275.5×10³ mm³

The lightest S-Shape I-beam using appendix Structural Shapes, is

     S250×38

However, now the problem needs to be solved again using the beam weight as an applied load. The new moment at the wall will be,

     M_w-2 = 10(2.333) + 2(9) + 4.5(0.038)9
           = 23.33 + 18 + 1.539 = 42.87 kN-m

The added moment, 1.539 kN-m, is small and should not make a difference in the beam size selection. The new Section Modulus will be

      S_min = M_w / σ_allow
              = 42.87 / 150,000
        = 285.8×10³ mm³

The lightest beam is still S250×38