Long pause for thought, for which I thank you...
Don't forget to include the effects of lateral loading from sprinters accelerating out of the saddle. You've assumed that all forces are in plane.
Agreed. But it's not possible to quantify lateral loads AFAIK. As a fixed wheel & tandem rider I am aware that riding out of the saddle is a very complex dynamic balancing act. As a result of my riding experience I am inclined to share Brandt's opinion that lateral forces are not critical. I suspect that the (secondary, in IC engine jargon) out of balance effect of legs at a cadence above 180rpm, with a seated rider, may be bigger. I'll listen if some empirical evidence turns up to show otherwise.
You suspect that the spokes have failed from fatigue. Tandems are following the solo bike trend towards stiffer rims and fewer spokes. 36 spokes are demonstrably holding up for both road and track teams. Most wheel builders use the same spoke tension for 28, 32 or 36 spokes if the rim is able to stand the resultant compression. If the spoke fatigue problem is the result of an insufficiently supported spoke elbow (as seems likely), how will 4 more spokes fix it?
I don't think I understand what you're saying that about insufficiently supported spoke elbow. I'm keeping an open mind.
Meanwhile some basics. Fatigue failure is fatigue failure. It's caused by cycles of repetitive stress that are lower than the stresses that would cause immediate failure. Preventing failure in the egineering world is usually based on reducing stresses, though sometimes specifying better materials or manufacturing techniques (ever heard of nitrided crankshafts for car engines?) may work. Undergraduate text books explain it better than I can.
The wheel in question has had about 5 million load cycles (in this case rotations). S-N data for steels shows a fatigue limit (the stress at which fatigue failure will never occur) is reached after 50 million cycles and this typical fatigue limit is only about 5% lower than the stress for failure at 5 million cycles.
So, increasing spoke count by 11%, which, at my level of naivety in stress analysis should reduce spoke stresses by a tad more, may well be enough to reduce stresses below the fatigue limit.
Oops - minor linguistic problem: if a 40 spoke hub & rim had been readily available, it might have solved the problem. I think my original question has already been answered, within the limitations of this forum.
Are you still with me? I'm not very good at explaining things to people I don't know, especially without the feedback that comes from face to face contact. S-N curves are commonplace stuff for Mechanical Engineers (which I am not), but there is more to come.
The problem with the fatigue failure stuff is that it doesn't tackle the complexities of the real life load cycles that bikes experience on roads. That's the point at which I resorted to empirical evidence. Son & I both have 2 bikes. 2 have symmetrical rear wheels, 2 have dished rear wheels. I'll fill in details if you think it will help to provide a solution to the original problem.