What would happened to wall stubs and osb during the construction under rain and heat condition? What is the normal time frame for building the house before the lumbers get damage? i.e. no more than 6 weeks? Thanks

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It is always nice to get it dried in before the weather hits, but some rain or other bad weather really won't hurt anything. The OSB should be Exposure 1 OSB (should be a stamp on it somewhere), which uses waterproof glue. If the framing and OSB are really wet it will need some time to dry-out after the roof is on.