Hello mike, it is a crappy day here and because I have nothing better
to do we will work on your project. First we need to get the component
of the loads in the direction perpendicular to the neutral axis of
your rectangular tubing. As you guessed it "helps" that the beam in on
an angle. Both loads need to be multiplied by the cosine of 35 degrees
or 0.82:
Point load = 300 x 0.82 = 246 lb.
Distributed load = 900 x 0.82 = 738 lb.
Each side of the staircase supports 1/2 of the point load and the
distributed load. Even though the distributed load is applied evenly
along the length of the beam there is very little difference between
that case and assuming that the entire load is applied at the center
of the span. Also, the formula for a point load in combination with an
evenly distributed load gets much more complicated. For our simple
(worst case) solution we will assume:
P(1) = 246 x 1/2 = 123 lb.
P(2) = 738 x 1/2 = 369 lb.
P(total) = 482 lb.
Deflection = (P x L^3) / (192 x E x I)
Where:
L = 166 in. and L^3 = 4,574,298 in.^3
E = 30,000,000 psi (a constant for any steel)
I = 21.6 in.^4 (your value of 11.1 is for the wrong axis)
D = (482 x 4,574,298) / (192 x 30,000,000 x 21.6) = 0.01 inch
Your design looks very good to me and you should feel no bounce.
Please ask for a clarification if you have any questions about what is shown here.
Good luck with your project, Redhoss |