Maximum bending moment

Of course there's a formula. That's what beam tables are for - too look up things like the maximum B.M. without having to work them out from scratch all the time.

Here is a set of typical beam tables:

http://www.awc.org/pdf/codes-standards/publications/design-aids/AWC-DA6-BeamFormulas-0710.pdf

Your beam is Figure 1, page 4.

BTW, the calculation of the total weight of the wall above has a small error. W = 0.145 MN, rather than 0.145 kN.

 

The problem is trying to figure out the dimensions of the cross section of the beam so that max. bending stress is limited to 7.5 MPa.

Remember, in bending, σ = M ⋅ y / I

It is postulated that the beam supporting the brick wall is twice as deep as it is wide, or w = b and d = 2 ⋅ b, and I = w ⋅ d³ / 12 and y = d/2