Old way (school way or double checking way) would be treating each part of the bike as a perfectly rigid, one-dimensional member. Creating a free body diagram (method of slices) at every joint, and solving for every load. You can then compare that to the deformation and stresses of every member/joint. The member/joint with the lowest factor of safety is your limiting load, if the FOS is lower than desired, you back calculate your load based on the desired FoS.
New way is just let <insert CAD program here> do finite element analysis for you.
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