How heavy does a roof need built to support a 185# / per square foot load?

The framing you describe sounds insufficient to support a "green" roof. Additionally, the reliability of the rubber membrane can be shaky; allowing ingress of water which could induce rot and deterioration of wood framing. For not too much more cost per square foot, I would suggest looking into metal form deck over light steel bar joists as a framing scheme.

I highly recommend contacting a local structural engineer to go over your house plans. He (or she) should be able to recommend an economical framing option. An architect would be able to point you in the right direction with regards to green roofing materials. I have heard of a light weight planting medium (kinda like the green water heavy foam used in floral arrangements) that can be used to greatly reduce the weight of the green roof and still sustain plant life.

I do not know where you are located, but if you respond I can suggest a couple of firms which might provide consultation on your project.

What is the span and the diameter of the logs? What kind of logs are they? What is the density of the straw bales?

With out this kind of info, know one can really answer your question.

My guess is the logs are spaced too far apart by a factor of two or three.

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