As per title, I would like the equation to calculate the effective failure rate of a system or branch of parallel/redundant components whose failure rates follow a Weibull distribution. I found what I need for components whose failure rate is described by an exponential distribution at pag. 394 (table 6.2.1-3, equation 1) of the book "system reliability toolkit" that can be found here. I would like to find the analogous for Weibull distribution.
RAC used to make their docs available for free; not so much any more. Here's a link to a previous edition: ReliabilityToolkit1993. Take a look around page 90 for some derivations of reliability equations. Put simply:
1) serial components In this setup, if any single component fails, the system fails. THerefore, the reliability is the product of the individual item's reliabilities. E.g., if you have three items with predicted failure rates of 99%, 95%, and 99%, the system value is 93.11 %
2) parallel components In the simplest case where you need one out of N working, calculate the probablity that all of them will fail and subtract from 1 to get the net reliability. For the same three items above, then, $P_f = 1 - (1-.99)*(1-.95)*(1-.99) = 9.99995e-01 $. Now, if you need to have, say N out of M identical parallel items running, you have to write out a slightly longer equation.
Nearly any system can be organized as a collection of series and parallel component operations, so the above 2 rules handle most situations. There are additional tweaks for spare components which are left running ('fully parallel') or spares which are left off until needed, presumably increasing their lifetime after being turned on, etc.
To answer your specific question, pretty much just place the Weibull function for each component where the exponentional function is in the document you referenced.
I believe there are some good handbooks available for download somewhere at MIL DITC , NASA, etc. as well, so a little websearching may be helpful.
ETA: there's a treasure trove of calculators at this site, along with detailed equations and explanations of each one.