In thinking about a weak matrix, just remember that it is the project manager that is weak and will be nothing more than a facilitator. The functional managers control everything. Even if you were to build relationships with the functional managers, you will find the employees are still loyal to them. And the employees will choose up sides by whatever person is most convenient for them at the moment. What's more, if there are too many managers to deal with, the communications paths become impossible.
There is only lose-lose in this environment and it has been well documented in many business books, and not just for project management. The best thing to do is nudge the organization in the direction of a balanced or strong matrix, if you can, or at least encourage the reduction of the number of managers you need to interface with.