Is it permissible to take monthly interest from the bank to cover family expenses, knowing that the principal amount may be lost if not invested in this manner?

If the bank is usurious (Riba-based), then the returns to the owner of the money deposited therein are forbidden interest (Riba). Covering family expenses does not permit this grave sin, for which Allah has threatened its perpetrator with war. The money owner is not compelled [to engage in Riba], and lawful ways of investing money are numerous. Allah Almighty has permitted all types of trade. Allah destroys Riba, so you must fear Allah and invest your money in what He has permitted, and stay away from Riba, which will bring destruction upon it.