With the beam oriented either vertically or flat, the calculation is simple. I use either Ix (strong axis MOI) or Iy (weak axis MOI) of the section and the equation d = 5wL^4/384EI. For the rotated section, though, I can think of 2 different ways to calculate the total deflection, which give very different results:

Method 1:

[ol A]

[li]Calculate the component of the load acting in the direction of the member's local y-axis. Then using this load (we'll call wy) and Ix, calculate the deflection, dy, which is the deflection in the direction of the member's local y-axis (strong axis).[/li]

[li]Using a similar procedure, calculate the deflection dx, in the direction of the member's local x-axis (weak axis).[/li]

[li]Calculate the total deflection: d = sqrt(dy*dy + dx*dx)[/li]

[/ol]

And Method 2:

[ol A]

[li]Given Ix and Iy, calculate I_rotated, which is the MOI with respect to the Global X-Axis, which is a horizontal line going through the centroid of the section (I should have included that in the sketch above). From my strength of materials text: I_rotated = (Ix+Iy)/2 + cos(2*angle)(Ix-Iy)/2[/li]

[li]With I_rotated calculated, calculate d from the equation above (using I_rotated in place of I).[/li]

[/ol]

I'll follow-up with a spreadsheet showing the large difference in values between the 2 methods.